Hollow Metallic and Dielectric Wave­guides for Long Distance Optical

Transmission and Lasers

By E . A. J. MARCATJ Ll and R. A. SCHM ELT ZE R

(l\hllllsc ript. received J une 12, 19(H )

'Phe field configurations and propagation constants of the normal modes are determined for ll. hollow circular waveguide made of dielectric material or metal for alJplication as an optical waveguide. The increase oj allentla­lion due to Cltrvlltllre oj the a:ris is also determined.

'Phe allClwation of each mode is found to be proportional to the square oj the free-space wavelellgth X a,Il(/ inversely proportional to the cllbe oj the cylinder radius a. For a hollmv dielectric waveguide made nf (Ilass wilh II = 1.50, A = J Po, and a. = J 111111 , all. attenuation, oj 1.85 db/ kill. is l)redicled fOT the minim.1.I.1n-loss mode, Ellll . Th is loss is doubled for a uuli'lI,S of curvature of lhe (twirle Q:I." ,:S Il ~ 10 km. Hence , diel.eclTic materials do unt sre'tn suitable for 'lise in hol-l.ow drcular wavegu.ides for lOll(l d'istance optical transmission beca'use oj the lri(lh loss inlrocl'tlcetl by even mild curvature oj Ihe gu.ide a:l'is. Nevertheless, dicleclTic materials are shown fo be vcr!! allf·active as guiding me(iia Jur gaseolls ampl'ijiers and osciUalors, ·not only because of the low atlenuation bllt also because the gain per unit length oj a dielectric tube containing H e-N e "masing" mixture at the r-i{fht pressure can be considerably enhanced by reducing the tube diameter. I n this applica­tion, a small guide raditls is desirable, thereby making the curvature of the g-u.icie axis not critical. ForA = 0.6328J.l and optimum radi'usa = 0.058 mm, a maximmn theoretical (lain of 7.6 db/ n/. is predicted.

It is shown thal the hollow metallic Ci1"CldaT waveguide is In.!' less sensitive to C'llTval'll1"e of lhe guide axi s. 'Th1:s ,is due to the c01nparatively lar{fe complex dielectric constant exhibited by metals at optical frequencies. For a wave­length A = lJ.l and a radius a = 0.25 nnl1, the attenuation for the minimum loss TEo! mode 't:n an al'u.minwn waveguide is only 1.8 rlb/ km.. This loss is doubled for a radius of cm·va.l1ll'e as short as R ~ 48 meters . For A = 311 and a = 0.6 '111.'111 , the alten'llal'ion of [h e TEo, mode is also J.8 db/ km,. 'rhe uwi'lts of cUTval'ltre which doubles this loss i s appro:l'imately 75 meters. 'l'he

1783

1784 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1004

straight guide los8 for Ihe E HII mode for A = 11' and a = 0.115 mm is 57 db/ k1ll and i8 increased to 3110 db/ bn for A = 31' and a = 0.6111111.

I n view of lhe low-l088 characleristic of Ihe TEO! mode in melallic wave­guides, the high-loss discrimination of noncircular electric modes, and the relative insensitivity to axis curvature, the hollow metaHic circular wave­guide appears to be very attractive as a transmission medium for long distance optical communication.

I. INTRODUCTION

During recent years the potentially large frequency range made avail­able to conununicatiolls by the development of the optical maser has stimulated much interest in efficient methods for long distance trans­mission of light. The most promising contenders for long distance optical transmission media consist of sequences of lenses 01' mirrors, highly reBective hollow metallic pipes, and dielectric waveguides. I •1O

In this paper we present an analysis of the field configurations and propagation constants of the normal modes in a hollow circular wave­guide which, because of its sin1plicity and low loss, may become an important competitor. The guiding stlUctW'e considered here may con­sist of an ordinary metallic pipe of precision bore whose inner surface is highly reflective, or of a hollow dielectric pipe - i.e., one in which the metal is replaced with dielectric. Although the transmission charac­teristics of metallic waveguides are well known for microwave fre­quencies, this theory is invalidated for operation at optical wavelengths, because the metal no longer acts as a good conductor but rather as a dielectric having a large dielectric constant. In t he subsequent analysis, therefore, both the dielectric and metallic guide are considered as special cases of a general bollow circular waveguide baving an external mediwn made of arbitrary isotropic material whose optical properties are charac­terized by a finite complex refractive index. If the free-space wavelength is much smaller than the internal radius of the tube, the energy propa­gates not in the external medium but essentially within the tube, bounc­ing at grazing angles against the wall. Consequently, there is little energy loss due to refraction. The refracted field is partially reflected by the external surface of the tube and may, in general, interfere constructively 01' destructively with the field inside the tube, decreasing 01' increasing the attenuation. Because of the difficulty of controlling the interference paths, it seems more convenient to eliminate the effect completely by introducing sufficient loss in the dielectric 01', in the case of a glass di· electric, by frosting the external surface. The field in the hole of the

LONG DTSTA NCE OPTICAL COMM UNICATION 1785

tube is then unaffected by wall thickness. We shall therefore simplify the analysis of the hollow circular waveguide by assuming infinite wall thickness, as depicted in Fig. I.

This strllcture will be shown to be attractive as a lowwloss transmission medium for long distance optical conullunication as well as fOI" optical gaseous amplifiers and oscillators . It is known, fOI" example, that in a tube containing a He-Ne mixture such thl1t the product of mdius and pressme is l"Oughly a constant, the gain per unit length is inversely proportional to the mdius of the tube. II On the other hand, we find in this paper that the attenuation of the normal modes is inversely pl"Opor­tional to the cube of the radius. Hence there is all optimum tube radius fOI" which the net gain pe l" uni t length is a maximum. Furthermore, because the guidance is continuous, there is no need fOI" periodic focusing. Consequently, no restriction need be imposed 011 the length of the amplifying or oscillating tube.

We begin by analyzing an idealized guide I",ving a stmight axis and a cylindrical wall. The results are then extended to include the effects of mild curvature of the guide axis by finding a perturbation correction for field configurations and propagation constants of the idealized straight guide.

II. MODAL ANALYSIS OF THE GENE nAL STHAIG I·I'I' CillCULAR WAVEGU IDE

Consider a waveguide consisting of a circular cylinder of radius a and freewspace dielectric constant fO embedded in another medium of dielecw tric 01" metal having a complex dielectric constant f. The magnetic permeability ~o is assumed to be that of free space for both media. We are interested in finding the field components of the normal modes of the waveguide and in determining the complex propagation constants of these modes.

The problem is substantially simplified if it is assumed that

.:,::: ... : ........ -; .. :: ....

Fig . I - Hollow dielectric waveguide.

T , , za , , , .L

(1 )

1786 THE nEI~L SYSTEM TECHNrCAL JOURNAL, J ULY 1064

and

1 ("Il k) - 11 « 1

where k = WVEo,UO = 27r/ X is the free-space propagation constant; 'u.n",

is the mth root of the equation JII~l (1Lllm) = 0, and 11, and m are integers that characterize the propagating mode; • = ~ is the complex refractive iudex of the external medium; and 'Y is the axial propagation constant of the mode under consideration. The first inequality states that the radius a is much la rger than the free-space wavelength X. In the case of metalization of the external medium, I II I may be quite large but is fin ite at optical frequencies. The second inequality restricts our analysis to low-loss modes, which are those whose propagation constants "I are nearly equal to that or rree space.

The field components of the natural modes of the most general circular cylindrical structure with arbitrary isotropic internal and external media have been determi ned by Stratton." T his structure supports three types of modes: first, transverse ci rcular electric modes whose only (ield com­ponents am E, , H rand II ~ ; second , transverse circular magnetic modes whose components arc H, , Er and Ez; and third, hybrid modes with all the electric and magnetic components present. The approx imate field components of these modes are written below, They have been derived using the inequalities (1) and neglecting terms with powers or Ala la"gcr than one, The superscripts i and e refer to the internal and external media, respectively,

I. Circular electric modes TEo. (11. = 0)

HrOm' exl' i( -yz - wI)

H~".i =

(2) ElOm

e = -1

H rome

i "var(V' _ 1) J O(lIom)

I-

." Eo -, v;;-=l -1'0

exp i[k. (,· - a) + -yz - wI] J

2. Circular Illagnetic modes 'Drom ( n = 0)

E,om' = .], (k,1-) 1 E" . "110m J ( ' ) 'zOm = 1 ,- 0 1',1'

,a exp i ( "(z - wi.)

Hoom' = V~ J, (I"r) J

. 'UOmJII(/ fll", ) . '/ . . 1 (' 1) ex p ' [k,(,.- a) + ,,/z - wll .,val' II -

3. Hybrid modes EH", (n '" 0)

E ... m' = [.1,_, (kir) + ~"',m' vV' - 1.1',(I.: ,,-) J ... 11. 1' (1.

. cos .,,(0 + eo)

E",.' = [.1 ,_, (k ir) + i~I,;;,' Vv' - 1 .1,(k,,·) J exp ihz -wt)

·sin n(e + eo)

~ U,m

i , . 1 ( " 1) .1,(u.m ) Ivar II'" -

E'.m' = cos n(e + 00)

E..m' = sin n(e + 00)

E .. m' = - Vv' - 1 sin n(O + 00 ) J ·exp i[k,(r - a) + ,,/z - wtl

I-

e 2 to I .. H'nm = II - Ern ... ~o

Ifr""." = 1-' 0 • - - E" .. ", ~o

1787

. (3)

(4)

1-------20 - ---1 I I 1!8\ I

I ,I I t I

~ Jt (3.e32 ~)-p-~ Jt (7 .0'6~)N~

TE Ot TE~

Jt(3.832~) !~1 I' TMo1 a

(bl CIRCULAR (a) CIRCULAR ELECTRIC MODES

_*. 'i ~,_ t t

.' .... . c":"., '" t I .<:,:,i"

J_ "(l l .065~)~~ J2(5 .'36~)I.C'>J ~ EH_12 EH3t

(C) HYBRID MODeS

J t (7.0I6%)p , r = a TMo2

MAGNETIC MODES

Fig. 2 - Electric field lines of modes in hollow dielectric waveguides: (n) circu­lar electric modes, (b) circular magnetic modes, (c) hybrid modes.

1788

LONG DISTANCE OPTICAL COl\fl\fUNICATION

where t he complex propagation constant')' satisfi es t he relationships

k/ = k2 _ ')'2

k~2 = y2e _ "/

and 'U"", is t he 'UI.th root of the equation

J "_I('U",,,) = o.

1789

(5 )

(0)

As usual, I 'It I is t he nUlliber of periods of each field component in t he 0 direction, a nd 111 is both the order of the root of (6 ) and t he number of maxirna and minima of each component counted in the radial direction within the internal medium. The constant 00 appearing in (4) will be­come of interest later on when we study the waveguide with curved axis, because it wi ll admit any orientation of t he transverse electric field relative to the plane of curvat ure of the guide axis.

For n. = 0, the modes are either transverse electric T Eo", (2), 0 1'

transverse magnet ic '1'1 10", ( :3 ) . The lines of electric field of t he T Eo", modes are transverse cOJlcentri c circles centered on the z axis. T he lines of magnet ic fi eld a re ill planes containing the z axis. Similarly, t he lines of magnetic fi eld of the Tl\Iom modes a re transverse concentric circles centered on the z axis with the electric fi eld contained in radia l planes. The electric fi eld lines of the modes TEu] I TEo2 I T i\ [ U] and Ti\102 are shoWJl in Figs. 2(a ) and 2(b ) ; each vector represents qualitatively the intensity and d i" ection of the local field .

For n ~ 0, the mod es are hybrid , E H"", (4 ) j therefore, t he magnetic and electric fi eld a re three-dimensiona l with relatively small axial fie ld components in t he internal medium. Thus t he hybrid modes are a lmost transverse.

Let us exam ine t he proj ection of these three-dimensional fi eld lines on planes perpendicular to the ax is z of the waveguide. The d ifTerential equations foJ' the proj ected li nes of electric field in both media are

(7)

Er"n/ as well as E, .. ",; contain byo terms as given in (4 ) . Both are neces­sary to satisfy t he boundary cond itions. If we neglect the second term , however, no substant ia l 0 1'1'01' is iutl'Oduced except very close (a few wavelengths ) to the boundary, where the second term dominates as

1790 THE BELL SYSTEM TECHNlCAL JOURNAL, JULY 1964

the first tends to zero. With this simplification, the differential equations (7) in both media become identical

(I / r)(drl d8) = tan n8.

Upon integrating, one obtains an equation for the locus of the projected electric field lines

(rl r,) " cos n8 = 1 (8)

where ro is a constant of integration that individualizes the member of the family of lines. The electric field of an EH,,", mode is different from that of ElI_nlll mode.

The projection of the magnetic fi eld lines is determined in a similar way. These equations are

(rl ro) " sin n8 = 1

fol' the intel'nal mediulll and

(rlr,) ".' sin n8 = 1

fo l' the external rnodium .

(9)

The PI'ojections of the internal electric (8 ) and magnetic (9) field lines are identical for any given mode except for a rotation of .. /( 2n) radians around the z axis. I n Fig. 2( c) the lines of the electric field in the internal llIedium are depicted for the first few hybrid modes. Again the vectors represent qualitativcly the field intensities and directions.

What happens at the boundary? Consider, for example, the projected electric lines of mode EHII , as shown in Fig. 3(a). These field lines satisfy (8), an equation which is valid everywhere except near the boundary. The boundary conditions are violated in Fig. 3(a) because there is continuity not only of the tangential electric component but also of the normal component. The internal normal component must be .' times larger than the external one. Consequently, the electric field line must be discontinuous. This result is shown qualitatively in Fig. 3(b).

A three-dimensional representation of the field lines is far more com­plicated than thc two-dimensional one depicted in Fig. 2. As a typical example, the electric field lines of the EH" mode are shown in Fig. 4 in a three-dimensional perspective.

The propagation constants of the TE,", , TM,", and EH,,", (n r' 0 ) modes are determined below (21). It is found that the hybrid mode EH_ I " I.m is degenerate (same propagation constant ) with the EH I " I + 2.m ; i.e., for every hyhrid mode with negative azimuthal index there is a degenerate hybrid mode with positive aximuthal indcx. The

LONG DISTANCE OP'n CAL COMMUNICATION 1791

la) (b)

Fig. 3 - (a) E lect. l'i c fie ld lines IIf E B Jl mode v iuhtting buundn.ry condi t.iuns ; (b) same E H II Illude wi t h electri c field lines quali lat ive ly con celed.

transverse modes TEo ... and Ti\fo,,, and the hybrid modes EH,,,. and E H:!III have no degenerate cOlili terpart.

J I' the £-ield components of the degenerate EH_ I " I ,m and E I-I I ' j I +:!, m

modes (4) arc added, \\"e obtain new composite modes whose electric and magnetic fi eld lines project as straight lines on a pla ne perpendicular to t he z ax is. SOllie of those composite modes are shown in Fig. 5.

It should be noted t hat if the l'efractive index of the extel'llal medium,

F IG. ·1- Culnwll.y view of elecl,ri c fie ld lines of E I:[ !~ mode. The ax ia.l period is grossly exnggemled.

1792 THE BELL SYSTEM TECHNICAL JO URNAL, JULy 1964

J_3(9.761~) ~ 'I'

d

EH-21 • EH41 EH _u + EH42

Fig. 5 - Electric field lines of composit.e modes EH_1"1,,., + EH1 1I1+t,m .

II, is very close to unity, then for each value of m, the TEom , TMom and EH'm modes also become degenerate (17 ), (21) and the sum of the components of TEo", (2) and EH'm (4) yields a new composite mOOe, as shown in Fig. 6. This mode, together with those in Fig. 5 and the EH'm of Fig. 2(c), form a complete set that closely resembles the set found for interferometers with plane circular mirrors or for sequences of circular irises. I

Let us now consider the field intensity distribution outside and inside the hollow dielectric waveguide. The external field ( 2), (3) and (4) has the radial dependence

exp {ik,( r - a)]

vr

LO NG DlS ']'AN(' E OPTI CA L COMro.1UN ICA']'ION 179~

J'(3 ' 83 2 ~)~ , r

d

Fig. G - Electric fie ld lines of ('ompos ilc modes TEo ... + ER~m .

From (5) a nd (20) we obtain, ncglecting terms of order (A/a )' and higher, Ire = kv' ,,2 - 1. The rad ial dependence is then

exp [i/o V;;-=-](,· - a) 1 -v';

If the dielectric is lossy, t he refractive index v has a positive imagina ry part. The external electric and magnetic fi elds then oscillate with period of the order of A/ I V- - 1 I and decay exponent ia lly in the radial direction. The maximum ficld intensities in the external medium OCClir at the boundary r = a .. Being proportional to X/ a, these maxima are small.

The fi eld intensity inside the hollow waveguide is more interesting. Again if we substitute 'Y (20 ) into (2), (3) a od (4) and neglect terms of the order X/ a, only the internal transvcrse components remain.

For TEom modes

EtJOm ;

For TM, ... modes,

. ;;, If .. .' 11 ;'

( 10)

( 11 )

1794 THE BELL sYs'rEM TECHNICAL JOURNAL, JULY 1904

For ERn ... modes,

E,,,,,,' = -/~ /-l ,,, .. i = J"_1 (U""'~) cos nO

E" .... i = 1- () ,uo i l' . ~ H olI", = J II _ 1 '/.t il 1/! a sm nO.

( 12)

The field con1ponents of each mode have approximately the samc radial dependence, varying as Bessel functions of the first kind, and tending to negligibly small values at t he houndary (6). This approximate radial dependence (10), ( I J) and (12) is reproduced undcr each mode pattern in Figs. 2(a), 2(b) and 2(c) .

III. PUOPAGATION CONS'I'A NTS FOR THE GENERAL CIRCULAR CYLLNU1HCA L

GUIDE

In this section wc shall determine the propagation constants 'Y, of the TEo .. , TMo .. and EH ... modes in the straight hollow guide at optical wavelengths. The propagation constants are the roots of the following characteristic equation for the general circular cylindrical structure. a They are related to k, and k, by expressions (5 ) .

[J,: (k,a) /.; , H" (Il' (k,.a) J[J,: (k,a) ./"(";") - C H .(Il( !.:,.a ) ./ ,,( k;a )

v'k, /-I" (1 )' (k,a)J - To H ,, (I) (k,a )

= nA I_ ~' ~ [ J' [ ()J" kkia. ktl

( 13)

This Cfluation is s implified substantially when the approximations in ( l ) are introduced. ,' illce !.' ,lI » I, the asy mptotic value of t he Hankel run ctions may be used

Since

f/ ,, (I)' (k,a) . -;-;"-;~-"-+ ~ , + O( 1 I k,lI ) , H ,,(Il(k,lI )

/r,n » I.

:!.- ~ v' (~) « J k,a ( v' - 1)1 2,..a

(14)

( 15)

powers of v' l k,a la rge>' than one sha ll be neglected. The characteristic equation then simplifies to

.I,,_,( !.:,lI ) = iv,,(k ;fk ).I, (k,a ) ( 16)

where

LONG DISTA NCE OPTICAL COMMUNIC~\ 'I'ION

Vv' - 1

Vv' -Hv' + 1)

Vv' - 1

for TEo" modes (n = 0)

for TMo", modes (n = 0)

for EI-l,,,,, modes (n ¢ 0) .

1795

( 1 i )

To solve the characteristic equation for k.-a we notice that because of (1) and (5), the right-hand side of (10 ) is close to zero. Using a perturbation techni'lue and kecping only the first term of the perturha­tion,

kia ;:::::::: It ,, ,,.(1 - ill,J ka) (18 )

where 'Un". as before is the mth root of the eq uation

(19 )

The validity of (18) is assured provided that the order of the mode is low enough so that i"" I 'nUIII « 1m. The propagatioll constants 'Y can then be obtaiued from (5)

_ ! (II""'X)' (1 _ iv"X )] . 2 2iTa iTa

(20)

The phase constant and attenuation constant of each mode are the real and imaginary parts of 'Y, respectively,

~"m = Re h) = 2; { I - H';";~ J[ 1 + [m (:": )]} (21 )

lIn (~ ) - ("""') ' X' R ( ) an", I - 2iT a:! e II" .

IV. PROPAGATION CONs'rANTS Fan S'rnArGHT DIELEC'J'RIC GUIDES

For guides made of dielectric material , 11 11 is usually real and inde­pendent of A, so that the phase and attenuation constants are

1796 THE BELL SYSTEM TECHNrCA1~ JOURNAL, JULY 19M

= 2 .. {l _ ! (U.mX)'} A 2 2 .. a

1 for T E'm modes (n = 0)

v' 1 '

= (u"m)' ~ , (22) v-

for TM'm modes (n = 0) a ll'" 2". a3 Vv'- 1 '

!(v' + I ) for EH,m modes (n '" 0) Vv' - l'

The phase constant of modes in hollow dielectric waveguides have the same frequency dependence as modes in perfectly conducting metallic waveguides when operating far from cutoff; hoth tl'ansrnission media arc then similarly dispersive.

The attenuation constants are proportional to X2/a3. Consequently,

the losses can be made arbitrarily small by choosing the radius of the tube a sufficiently large relative to the wavelength X.

The refractive index v affects the attenuation of each of the three types of modes (22 ) in different ways. This fact is reasonable on physical grounds. TEom modes can he considered to be composed of plane wave­lets, each impinging at grazing angle on the interrace between the two media with polarization perpendicular to the plane of incidence. It is known from the laws of refraction that the larger the value of II, the smaller the refracted power.

TlVlo"l modes may also be thought of as consisting of plane wavelets, but with the electric field of each now contained in the plane of incidence. For v very close to unity, there is little refl ection and the refracted loss is high ; as the value of v is allowed to become large, each wavelet gets close to the Brewstel' angle of incidence and again the refracted loss is high. The minimum occurs for. = 0.

EH"m modes are composed of both types of plane wavelets. Therefore, ns is reasonable from the above argument, the attenuation constant a llm has a II dependence which is an average of those of TEom and TIVlom

modes. The value of II that minimizes all '" is " = .y'3 = 1.73. T he attenuation constants (22) are proportional to " .. m'. Some values

of " om (19 ) are presented in Table 1. For a fixed value of n the attenua­tion constant increases with In. This statement is not true for m fixed and n variable.

Comparing the attenuation constants (22) of the different modes, we find that the mode with lowest attenuation is TEOI if • > 2.02 and EHII if v < 2.02. Most glasses have a refractive index v ~ 1.5, and

LONG OISTANCE OPTICAL CO MM UN ICATJON 1797

TAllLE 1- SOto.'I E VALUES Ol<~ 'll"wl

II/ m I , 3 • 1 2.405 5.52 8.054 11. i9G

2 or' 0 3.832 7.016 10 .173 13.324 3 U I' - 1 5 .136 8A 17 I1.G2 14.79G ., or - 2 G.380 9.761 13.015 16.223

consequently for hollow glass tube EH II should be preferred . The attenu­ation of this mode (8686<>11 in db/ km ) has been plotted in F ig. 7 as a function of Aja for v = 1.50 usi ng A as a parameter. Typically, for a wavelength A = IJ.l and radius a. = 1 nun, the attenuation of the EHII mode is J .85 db/ km ( ~3 db/ mile) . If the radius of the guide is doubled, the attenuation is reduced to 0.2:H db/ km.

0

2 0

'? < or

...=>. w o o , I W ~

o o. z o ~ ~ o. w ~ ~ < o.

0.0

0.0

0.0

0

5

2

,

5

2

I

5

2

,

, }. = IO-~M 10- !>M to- 'M

\

1\ !\ \ \ \ \ \ \ \

, to-7M

1\ \ \ \ \

1\ \ \ \ \ \ \ \

\ \ \ \ \ \ \ \ \ \

\ 1\ 0.0 5 0 .02 0.01 0 .005 0 .002 0.001 0.0005 0 .0002 0 .0001

'fa

Fig. 7 - Ailcllunt.ion of E H" IWldes (1.85 A~/U 3) venms wnve lcngth/ radius (, ~ 1.5).

1798 1'HE DELL $YS'l'EM 'rECHNICA L JO URNAL, J ULY 1964

V. HOLLOW J)tELE(."l'Rl C WAVEGUIDE FOR OPTICAL MASER AMPLIFIERS

AND OSCILLATOnS

A mode traveling in a hollow dielectric wavegu ide fiUed with "masing" material experiences a net gain which is given by the difference between the amplification due to the active medium and the loss due to leakage through the walls. It has been shown" that in a tube filled with t he right mixture of He and Neat the propel' pressure, the gain G is in­versely proportional to the radius a of tbe tube. Then

G = ( A / a) db/ m (23)

where the radius a is measured in meters and the constant A is

it = 0.00056 db.

On the other hand, we have found that the transmission loss of the ER" mode in the hollow waveguide with a refractive index v = 1.50 is L = 8.685"" . F rom (22)

L = B( 'A' / a' ) db/ m (24)

where the constant B is

B = 1.85 db.

The uet gain per uuit length is then

G - L = ( /l / a ) - B ('A' / a')

pass ing through a Il Jaximulll at the value of the rad ius for which

a(G - L )/oa = O.

The optimulIl rad ius and the IlJaximulll llet gain are respectively

a.", = ~ 'A = 9 1.7X

(G -2 A ! \ 10- 6

L)m~ = 31 BI X = 4.8\ T db/ m.

1"01' t he He-Ne mixture, 'A = 0.6328 10-' m. Consequently

Q"pl = 0.058 mm

(G - L )", .. = 7.5 db/ m.

(25)

(25)

(27 )

Although the diameter of the tube is quite small , the gain pel' unit length is sufficiently large as to make hollow dielectric amplifiers and oscillators attractive for e:xperimentatioD.

LONG DIS TA NCE O IYl'IC.-\.L CO?IM UN ICATION 1799

Present-day confocal He-Xc mascI'S employ tubes whose approximate length and radius UrC 1 III and a nll ll respcctively . The gain per passage (23 ) is 0.22 db (",",j pel' cen t ) . lI' a hollow dieleetl'ic waveguide with an optimulll radius 0.058 mill were used , the sallie gain would be achieved with a length of only 0.22/ 7.G = 29 mill. This presents an excellent possibility for a very cOITlpact mnsc l' .

Even for radii largcr thall the optimum, the hollow dielectric wave­guide is still attractive. For exatnple with a = 0.25 111m, the gain is 2.6 db/ m, a value far larger than the gain 0.22 db/ m obtained for the 3-mm radius tube commonly used for rl1asers.

N evertheless, for long-wavelength masers the optimtlln values (26 ) are not pl'actical. COllsider fo l' example a tube containing an active material which amplifi es at A = 10- 4

111. Let us assume that the constan t il is st ill 0.00066. Then from (26), the opt imum radius and maxImum gain are

( r: - L )",", = 0.048 1 db/ lII. (28)

The gain is very small. It could be enhanced by rcducillg the mdius and by illcreas ing the refractive illdex "of tbe walls to a value much larger than 1.5. This can be accompli :-;hed if lIletal is used instead of dielectric, as is show n in the next section.

V I. A'L"I'ENUA'rI ON CONSTANTS Fon THE S1'RAIG I-I T METALLIC G UIDE

I n order to discuss the attenuation characteristics of metallic wave­guides, we shall need to have some quantitative information about the behav ior of metals at optical frequencies. \-Ve examine as a typical example the optical propcrties of aluminum, even though this may not be the most suit.able metal. The dispersion characteristics of the con­ductivity and relative dielectric constants of alu minulll have been studied extensively by Hodgson,i3 Beattie and Conn ,I'" and Schulz.H'

The data used below have been taken from a compilation of the resul ts of these studies,i6 and is presented graphically in Fig. 8. It is evident from these dispers ion curves that the dielectric constant for aluminum is much larger t.han fol' ordiliRry dielectrics and increases monotonically with \\"Uvelength in the range O .:l~ < A < ,1 .0~ .

The circular electr ic Illodes lu\\'c the lowest loss in IllctaUic wave­guides, while the circular magnetic and hybrid modes arc rapidly attenuated even for a wavelength as short as O.~~. T he attenuation con­stant nO! for the lowest-loss TEO! mode is plotted in Fig. 9 for wave-

1800 T I:IE DELL, YSTEM TECHNlCAL JO URNAL , JULY 1964

0 .0 • o. • o. 2

o. • • 2

• 'N ....=. 10

" Cf 20

'0

'00

200

'00

1000 0.'

1\

'\ ~

1\

'\ '\

1",,- 1\

"'" "--R. {.,~ r-.. ~m(V2)

r--.. "-

"'" '" ~'" ,"'-' 0.2 0.4 0.8 0.8 I 2 4

WAVELENGTH, X, IN MICRONS

F ig . 8 - D ispe rsion curve (or aluminum .. 1 = f i fO = Re ( .. ' ) + i 1m ( .. I) versus wavelength X~).

lengths in the range 0.31' < X < 4.01' for a = 0.25 nun, 0.50 mm and 1 mm. These data show a considerable improvement over that cor­responding to the lowest-loss mode ER" for the dielectric gu ide. We saw that for a hollow glass dielectric waveguide, the EH" mode has a loss of l.8 db/ km for a radius a = I mm and wavelength X = II'. The attenuation for the TEO! mode for the aluminum guide with the same radius and wavelength is only 0.028 db/ km. For a wavelength X = 11' and a radius a = 0.25 min, the millimwn-Ioss TEol mode for the alumi­num waveguide has an attenuation constant aOI = 1.8 db/ km . The same attenuation is achieved for X = 31' and a = 0.6 mm. The attenua­tion constant for the TE", mode under the last two conditions is a 02 =

6.05 db/ km. For a wavelength X = II' and a = 0.25 mm, the straight guide losses for the TMOI and EH" modes are approximately 145 db/ km and 57 db/ km, respectively.

VII. FIELD CONFIGURATION AND ATTEN UATION OF MODES IN THE CURVED

GUIDE

In order to achieve a more realistic evaluation of the hollow circular waveguide for long distance optical transmission, it is necessary to

LON(T DISTA N('E OPT ICA L ('O~Il\1UN I CATION

50

~ 20

10

'"i' {

5"", 2

~

"" m ~ w Q

o I

1: 0 .5

• UJ ~ 0.2

o. z I

o ~ 0 .05 ~ Z ~ 0 .02 I(

0 .0 I

0 .005

0.002

0 .00 I 4

'" ~ ~ --

~

2

-

'\. ~

,,\ r- :~~0. 25"''''

'\. " ~ '\. ,

I

~ I " I 0 .8 0.6 0.4

WAVELENGTH, >., IN MICRONS

0 .2

1801

0.1

Fig. !) - At.lenu:~t. i on of TEol Ill ude , a o! versus wnvclengt.h }..(p), for a lumin ulll guide.

evaluate the efTects of mild clln'atul'C of thc guidc axis. This is most easi ly accomplished by determining II perturbation cOlTcction for both the field con figllrat ioll and t he attenuation constants for t he idealized straight guide whosc chamctcl'ist ics ha\'c heell described above.

V III. FOItM U LATION OF THE PHODLEM

Consider t.he toroida l system (1", 0, z) wit h metric coeffi cients

e,

,. (29)

c, I + ,./ R sin 0

as depicted in Fig. 10. In t hi s system of coordinates, a differentia l length is given by

ds = (r,'d,.' + <,'dO' + r,'dz' )l ( :30 )

where I? is t he radius of cllrvature of t he toroidal system and is chosen

1802 'I'HE BELL SYS1'EM TECHN ICA L JOURNAL, J ULY 1964

Fig. 10 - The curved hollow dielectric waveguide IUld the lIlisocinted toroidul coordinatc systcm (", 0, Z ).

equal to the radius of curvature of the guide axis, so that the guide wall is located at't = a, and the axis of the guide coincides with the curvcd z-axis. In this toroidal coordinate system, ~Iaxwell's C<luations are

~ 1 (1 + "IR sin 0)3C.1 - i'Y,"X, + i",.,.( 1 + li R sin 0)&, = 0

i 'Y,3C, - i {( I + "I R sin O)JC,I + i",, ( 1 + li ll s in 0)&0 = 0 a ..

i ( .. 3C,) - -'!. x, + i"" .. &, a.. ao o

:0 I ( I + .. I R s in 0)&,1 - i'Y,6, - ,;"'w(1 + ,.IR s in O)X, 0

i'Y ,&, - i 1( 1 + liR s in 0)&, 1 - i",,,(1 + li R sinO)3c, = 0 ar

a () a . 0 !iT ,,6, - ao &, - ,,,,,,,,3C. =

whcre we have omitted the COllllllon factor

exp i ( 'Y.z - ",I)

in which 1'c is the propagation constant along the curved z-axis.

(a 1 )

'rhe toroidal system ("',0, z) and the CUl'ved waveguide degenerate into a cylindrical system a nd a stmight guide, respectively, as R ap­proaches infinity. :i\laxwell's equations for the straight guide are there­fore obtained from (a1) by lett illg R ~ "' .

LONG O I~T:\ NCE OI'T I CAL COM~IUN IC.'\T I O l'\

fe H, - ':'Y ,.H. + iw, rE, 0

i 'YH , - i H, + ·iw,}!.', 0 ar

i (rH.) - aio H, + iWfl'E, 0 ar a - E ao ' o

~ (rE.) - ~ F:, -iwwH, = 0 Br ao

1803

where 'Y i ~ the propagation COtlstant for the straight guide, and the superscript i and subscripts mn a l'(~ supprcssed.

IX . SOL UT ION I;'Q H T HE CURV E D GU IDE

We proceed to solve (:11 ) for t hc fi eld vectors e, JC and obtain the pl'Opagation CO llstant 'YG for the curved guide as functions of the fie ld vectors y,;, 7l a nd t he propagation constant "1 of the st ra ight guide. T he latter quanti t ips a re known 1(2 ), (:1), (-1 ) a nd (20 ) 1. We introduce a para lllctcr

The range of ill tel'cst is that 1'01' whi ch the radius of curvature Il is so la rge t hat u « I.

Using a fi rst-ordcr perturbatioll technique, the solution of (3 1) is

0. = ( I + ur/ a s in 0) 1::.

0, = ( I + ur/ a sin 0)8 ,

0, = ( I + ur/ a sin 0) 11', + (-iu/ ka )( E, sin 0 + E. cos 0)

JC. = (I + ur/ a sin 0) 1-1.

JC, = ( I + ur/ a. sin O) H,

JC, = (I + ur/ a sin 0)1-1 , + (-iu/ ka )( H, s in 0 + H. cos 0).

(34)

The etl'ect of curvature of the guide ax is is to make unsy mmetrica l the

1804 T HE BELL S YSTEM TECHN ICA L JO UHNAL, J ULY 1964

transverse field configuration of the straight guidc. Each transverse component is enhanced in the half cross section farthest from the center of curvature.

To a first-order perturbatioll of (J , the pl'Opagatioll constallts of the curved and straight guide arc identical ; i.e. , 'Yc ~ 'Y. Kevertheless, know­ing the field components of the mildly curved structure, it is possible to calculate its attenuation constants a,,", (R ) = He 'Y, .

X . A'!"rEN UATION CONSTANTS a"m (R )

The mean radial power flowing into the dielectric pel' unit length at the surface of the guide is

I 1" p , = - He [e.JC, * - s,x, *J [I + ai R sin OJa dO. 2 0 r_ (I

(35)

The power How ill the axial z direction within the internal mediulll r < a is

1 [" 1" p , = 2 Jo

c Re [S,X,* - e.x, *J I' dO ill' (36)

and decreases along z at a rate equal to the radial flow pel' uni t length P, j i.e. ,

ell>, ? ( /') 1' P -1- = - .... a,,'" t. ~ = - ,

" (37)

where a"", (Il) is the attenuation constant of the mode under considera­tion for the curved hollow dielectric waveguide. Consequently

a"",( R ) = H P,/ P,) . (38 )

To compute P, we substitute the known fi eld quantities into (35 ) . This yields

P, = Re ,y~ ~~;~~~":)I r [ 1 + ~ sin 0 [' ( 1 + ai R sin 0)

{ ) } dO

... sin' n(O + 00 ) + cos' n (O + 0,)

for TEo"1 modes for TMom modes for EH"", modes .

(39)

Terms with powers of X!(2",a ) larger than two have been neglected . Upon integrating,

P _ R I ;;; 'It" ",' } .'('It"",) , - 11' 0 - ~ _~

J1.1J k-av ,,2 - 1

LONG m:-i'rANCE OPT ICAL COi\IM UNICATION

(J + 1U ' )

/( I +tu')

for TEo", modes

for TMom modes

~ (v' + I) [I + ~ u' (I + 0,,( ±1) -: - 1 2 2 2 v- + 1

for EH" ", mode:-i

where

'COS 200)J

0,,( ±1 ) = {~ : n = ± I

n '" ±l.

(40)

(41 )

The power P;t flowing radially ill thf' gll ide is ohtained by substituting (:14) into (:lG) and integrating

_ 1ra" , / ;; 'C. ) r u' [ ? ( . 9) / 'I} P z ~ 2 V ; 0 J" tt,,,,, \ 1 + i3 1 + ""n n -... 'U1l1ll . (42)

Hence

{ 4 ( 21ra )' ( a)' a"",( R ) = a"", ( "' ) 1 + 3- - , R-o 'U" ,,,I\ ,

[

I _ n(n - 2) + ~ ° ( 1) Re V~ ?" ] } .. I " ± ~+ lcos_vO . It - ~ p.

' m Re V " 1 .--where 0'"",( cc) = 0'",,, is the attenuation constant for modes in the straight guide ( ll = -n ) given by ( 21 ) . The attenuation eonstant a,,,,, ( R ) can also he written ill the fo llowing form

",,,,, ( R ) = a"",( "') + (a' / X'R") Re V"",( . ) (44 )

w h ere

1

V.' - 1

4 .' C)' 3 V.' - 1 'I~n~' H.' + I )

(45)

V.' -

{ n(n - 2) :3 (v' - I) )

. 1 - . , + -, 0,.( ± I ) ' + 1 cos 200 > . U,,'" ~ P J

1806 THE BELL SYS'I'EM 'I'ECHN I CAL JOU HNA I" JU LY 1964

The values of ReV"", ( " ) are always posit ive. Some of them have been calculated in Table II for a refractive i ndex " = I .SO.

The attenuation constant of any mode consists of two terllls (44 ) . Thc first coi ncides with that of the straight guide and is pl'Oportional to lI. " ,,/A~/aa; the second term represents an incl'ease in attenuation due to CUl'vature of the guide axis and is proportional to a3/ A'R,2u,tlmZ. There­fore the lowcr the straight guide attenuation constant ( small 'li. tlm'!A'/ a3

) ,

t he la rger the loss due to bends a nd vice verBa. From (43 ) 01' (45 ) we find that only for t he EH±I .• modes, t he orientation of the field wi th respect to the plane of curvature influences t he attenuation. If 60 = 0, t he electric fi eld in the center of the guide is in the plane of curvature and the attenuation is a max imum . 1"01' 80 = ±1I" / 2, the electric field is normal to the plane of cu rvature and the attenuation is a minimum . The ratio of maximum to minimum is mild , however. For the lowest attenuation mode E H II and" = 1.50, it is

If I v I » I , t hat ratio is

1' .. ", (60 = 0) _ 1 (" \f ( / ) - . ID.

"Ill 80 - 11" 2

V ... (60 = 0) =46 V "",(0 = .. / 2) ..

(46)

F rom equation (4:1 ) we find t hat t he radius of curvature which doubles the straight guide attenuation is

2 (2")'a' Ro = V.3 'U"II' )!

' [ 1 _ n(n - " 2) + ~ o,,( ± I) '11."", - 4. Re

Re v';;;-=-j" ]1 v' + 1 cos260 •

v'';- l

(47)

This value of Ro is only approxi mate since (43 ) was derived by assum­ing u « I.

XL EFFECT OF CURVA'['UnE ON ATTENUA'r ION OF MODES I N 'rHE HOLLOW

OI ELEC'r RI C WAVEGUI DE

For a straight hollow glass waveguide with v = l.5 and a radius a = I nun operating typically at a wavelength A = II' , the attenuation of t he lowest- loss mode EHu is au = l.85 db/ kill. Th is loss is doubled for a radius of curvatul'e Ro ~ to kill . Fa)' long distance optical trans­mission a radius of cUl'vature of at least a few hundred meters would

L ON(; I)I STA~('E O P'l'IC.'\L ('Oi\l~'I UN I('ATIOi'\ 1807

"1,,1 I , J , - I 2.57 (1 + 0.326 1.034 (I + 0.301 0.553 (I + 0.295 O.3..t7 ( I + O.2!J3

cos 280) cu~ 200 } eos 200) e08 280}

I TE 3.22 0.955 0 .455 0 .2M

0 'I'M 7 .22 2. 1-15 1. 022 0.50u

I 15.5 (I + O.24H 2.110 ( I + 0.279 1.034 (I + 0.284 0.554 (I + 0.286 cns 28 0 ) cos 28 ,,) cos 280 ) ens 2O,, )

2 5.22 1. 55 0 .735 0.432

:1 2.57 1.0:14 0.553 0.347

401' - 2 1.51 0.737 0 .430 0.287

he tolerable. Therefore hollow dielectric waveguides do not seem suita­bl e for long distance opt ical transmissioll .

0 11 t he other hand , the CUlyaturc i ll hollow dielectr ic waveguides for application in gaseous amplifiers a nd oscillators is not critical. For example l if a = 0.2;') llllll alld A = IJ..l. , the straight guide atteuuation is 0. 12 db/ llleter. The radius of curvatu re which doubles this qua ntity 1'01' the lossiest polarizatioll - i.e., ,,·ith the electric fi eld at t he centel' of t he guide cont.aincd ill t he plane of cUl'vature - is approximately i;'}O IIlcte l'S, a. "aluc ,,·ell wit.hi u the linlits of laboratory precision . COII­

S(-'qllc ll t ly, t he hollow dielectric) wan'guidc docs remain very attractive as a guiding IIlc<iiuli l fo r optical amplifiers a nd oscillators ,,-here a slIllLlI gu ide radius is desirahle, therehy maki ng the guide less sensitive to curvature of the ax is.

XII. E FVECT OF ('u nVATURE ON AT' I'E:\ UAT ION or M o nES I N THE M E'I'A I~ I, I C

GUIDE

The attelluatiOlI constallts a u". ( R ) for the lowest-loss in the curved metall ic guide are gi,-en by

ao",(R) ~ao",(") {I + ~G::)' (~)'}

TEo ... modes

(48)

whel'e an ... ( eo) is the attelluatio ll constant for the T I!:om mode in the stl'a ight gu ide, R = :c. For a radius u. = 0.25 mill and wavelength A = I J..l., the straight gu ide loss for the lo,,-est-loss TEol llIode, aOI ( 00) =

1.8 db/ km, is douhled for a radius of cur\"!lture of only Ro ~ ..18 meters.

1808 THE BELL SYS'l'EM TECHN ICA L JOU UNAL , J UL Y 1964

For X :l~ and a = 0.6 III III , the straight TEm loss is also 1.8 db/ kill a nd the radi us of curvature that doubles that loss is 75 m.

XIII. CONCL SIONS

The hollow dielectric waveguide at optical wavelengths supports a complete set of normal modes that are either circular electric, ci rcula r magnetic 01' hybrid. They resemble the modes found in a sequence of circular irises not only in field configuration but also in loss discrimina­tion among them . . Fo!' hollow rnetallic waveguides the mode discrimina­tion is far larger.

The field configuration and propagatioll constants have been de­termi ned. The attenuation is practically independent of the loss tangent of the dielectric but depends essentially on the refraction mechanisrn at the wal l. Assuming refractive index of the dielectric, 1.5 fo r hollow dielectric waveguides, the ERII mode exhibits the lowest power attenua­t ion , viz., 1.85 ( X' / a') db/ m. For a wavelength X = l~ and a tube radius a = 1 mm, the attenuation is only 1.85 db/ knl.

The hollow dielectric waveguide does not, however, seem suitable for long distance optical t ransmission because of the high loss introduced by even mild cUl'vatu re of the guide axis. Nevertheless it remains very attractivc as a guidi ng Illcdium foJ' optical amplifiers and oscillators, since here a small radius of the guide is desirable. Conseq uently, cUl'va­tUre of the guide axis is not critical. Filled with "masing" material, the hollow dielectric waveguide provides not only guidance but also gain which is almost inversely proportional to the radius. For the right He-Ne mixture, the maximum theoretical gain attainable is 7.6 db/ In provided that the radius is 0.058 mm. But even if the radius is 0.2.5 mm, the predicted gain is sti ll large, viz. 2.6 db/ Ill.

The metallic waveguide is superior to the hollow dielectric waveguide for usc in long distance opt ical transmission. Because of the relatively large dielectric constant exhibited by aluminum at optical frequencies, the attenuation constant for the lowest-loss mode TEO! is comparatively small and less sensitive to curvature of the guide axis. Fa!' a radius a = 0.25 mm and a wavelength A = 1p, the attenuation constant fol' TEO! modes in the straight aluminum guide is only 1.8 db/ km, which is doubled for a radius of curvature of about 48 meters. For a = 0.6 mm and X = 3~ , the TE" straight guide loss is also 1.8 db/ km but i doubled if the radius of curvature of the waveguide axis is 75m.

We have considered some of the theoretical problems of the hollow dielectric 01' metallic waveguide. The results are promising, Neverthe­less, tbe usefulness of these guides bas yet to be pl'oven experimentally,

LO NG IHSTANCE OPTICA L COMMUN ICATIO N 1809

and furth ermore t he attenuation constants discussed here do not include scattering losses due to surface imperfectiolls.

XIV. ACKNOWLEDG MEN'rs

It is a pleasure to thank R Kompfner and S. E. MilicI' for their suggestions.

REFERENCES

1. Fox , A. G., and Li , Tingye, Heson:lIlt Modes in a :Mnser In t.e rferometer, B.S .T.J ., 40, l\·rarch , 1961, p . 453.

2. Boyd , G. D., lind Gordon , J . P. , Confocal .Mult.imode Resona tor for Milli · mete r t.hrough Oplicnl Wllvele ngl.h IvIuscrs, B .S.T .J ., 40 , March, 1961, p . ~8n.

3. Boyd, G. D., a nd Kogeln ik , H. , Gencralized Confoca l Resonator T heory , B.S.T .J . , 41 , July, IDG2 , p . 1347 .

. J. Goubau, G ., and Schwering , F., On the Guided Propagation of Electrumag· ne t ic Wave Benms T rans. l.B .E ., AP-9, May 1961 , p. 2-18 .

5. Eaglesfield , C. C., Opt icnl P ipeline: A Tentat ive Asses!:lment, The Jnst,. of E lect . Engineers, J nll Ul\ry, 1002, p. 26.

n. Simon , J . C., nnd Spitz , E ., Prupugation Guidee de LUl1Iiere Coherente, .J. Phys. Hafliu llJ, 24 , Fehruary, 1963, p. 147 .

7. Goubuu , G., and Chri st,inn, J . H. , Sumc Aspects of Bcam Wa veguides fo r Long ni~lallf'e Transllli ~R i n ll n.l, Optical Fmquencies, l EEE Tl'llns. un Microwlwe T heury aud Techniques, MTT-12, M:trch , H)O.J, pp. D.S:r.J ., 212- 220.

8 . l'vl arc1l8c, D., and Miller, R. E., Annlysis of a Tubular Gus Lens, 8.S.T.,I. , this issue , p. 1750.

U. Be rrenllL n, D. \V. , A Lenl-l or Light. Guide Using Convect ively Disturtccl The r­mill Gmciient s ill Gn8cs , ll.~.T.J ., th is issue , .p. 1469.

to . Be rreman , D. W., A Gas Lens using Unlike, Counter-Flowing Gases, B.S.T .. J., t hi s issue , p. 1470 .

11. Gordon , E. I. , fl.lld Whi te, A . n. , Similar ity Laws for the He-Ne Cas Maser, Appl. Ph .v:;. Lette rs, 3 , December 1, 196.3 , p. 199.

12. Stra l.toll, J . A. , Elc('/.roll/(lYllelic Th eo"!I, McGraw-Hill Book Co , New York and London, lfJ4 I , p . 52·1.

13. Hodgson , J . N ., Proc. Ph.\"s.:::oc. (London ), B68, 1055, p. 593. 14. Beattie , J . H .. a nd Conn. G . K. T .. Ph i\. ("lag ., 7. 1055 , pp . . I6, 222, and 9S9. ]5. ~chul z , L. C., J . Opt. Suc. Am ., 41,1051, p. 10-17 ; 44,1954 p . 3Si . !G. Give ns, l\'1. Parke r . Opl,icnl Prnpe r l.ies of Metu! :;, Solid Stale P It Y·'1 i,·s , 6, Acad .

Pres:; Inc ., New York , HI5S, p. :n :\.