IEEE TRANSACTIONS ON EDUCATION, VOL. E-30, NO. 1, FEBRUARY 1987 43
Guided-Wave Modes in Cylindrical Optical FibersCHAI YEH, LIFE SENIOR MEMBER, IEEE
Abstract-This paper proposes a logical and simple approach to the Fortunately, in practical applications of optical fibers,mode designations in a cylindrical optical fiber. An important approx- the situation is not that helpless. A practical step-indeximation is introduced in the mathematical analysis to simplify the char-acteristic equation. Mode enumerations, field distributions, and prop- optical fiber is usually made of an optical fiber coreagation constants for some lower order modes are discussed and embedded in a cladding of slightly different refractive in-illustrated. The justification of using the linearly polarized mode des- dex. For best transmission, the refractive index differenceignation is illustrated. between these dielectric cylinders must be small [10]. Un-
der this condition, all fields within the guide becomeINTRODUCTION weakly guided. Taking advantage of this practical find-
ing, Snyder [11] and Gloge [12] approached this problemSateacher in microwave engineering, I am often by making An(= n, - n2), the differential refractive in-confronted with the problem of presenting the cor- dex, very small. They succeeded in simplifying the so-plex mode designations in cylindrical guided-wave opti lution considerably, thus making the mode designationcal fibers. Although there have been many books and pa- manageable.pers written on that subject, these materials are usually In this paper, we shall start by solving the wave equa-less suitable for classroom presentation. Many are very tions for a cylindrical optical fiber to obtain the generalcomprehensive, sophisticated, and theoretical, but the characteristic equation. The complexity of finding solu-physical picture is often masked by the multiplicity of tions of this equation for mode designation is pointed out.mathematical manipulations [1]-[4]. This causes a great The simplifications resulted from introducing An 0O isdeal of confusion when students are exposed to this sub- emphasized. A few simple lower order modes are usedject for the first time. It is the purpose of this paper to for illustration. We first discuss the mode designation,present a concise and logical picture of the subject while then the evaluation of the propagation constant, and fi-preserving its mathematical elegance, thus making it easy nally the field distributions for the lower order modes.to teach and easy to understand. Linearly polarized mode designation is introduced andThe propagation of electromagnetic waves along an justified. The discussion is limited to step-index fibers.
ideal dielectric circular cylinder of infinite length embed- Propagating modes, that is, modes that are guided alongded in an infinite and homogeneous dielectric medium has the fiber are emphasized. The radiating and evanescentbeen a classical problem ever since the turn of the cen- modes are discussed only relating to power loss in fiberstury. Rigorous solutions of the wave equations in cylin- [13], [14].drical coordinate system applied to this problem revealed It must be mentioned that this is a closed form analyt-that the field structure of this type of waveguide is much ical solution which is valid only for step-index wave-more complex than its hollow metallic waveguide coun- guides. However, practical multimode waveguides are aterpart [5]. In metallic waveguides, the propagating modes graded index for which no analytical solution exists incan easily be classified into TE, TM, or even TEM modes. general.Due to the nonconducting boundary condition of an op-tical fiber, all electric and magnetic field components ex- MODE DESIGNATIONist both within and without the boundary. The resulting Before we start to analyze the wave equations of a lightmode structure becomes extremely complex. It includes waveguide, we wish to describe the mode designation ofthe wanted propagating modes along the guide length and a fiber in words in order to prepare the readers for thethe unwanted radiating and evanescent modes elsewhere complex nature of the mode's structure. It has been rec-[6], [7]. All these modes are solutions of the general char- ognized [15] in microwave engineering that many propa-acteristic equations. Worst yet, all these modes can be gating modes can coexist in a hollow metallic waveguidecoupled to each other through any irregularities and non- or cavity. It is often necessary to identify the existingsymmetrics introduced along the fiber length [8], [9], [14], modes so that an appropriate mode for propagation or ex-[15], [22]. The study to distinguish these modes is, in- citation can be selected and all other unwanted modes bedeed, a formidable task. isolated or eliminated. In a cylindrical dielectric light
waveguide, the situation is more complex due to the ex-Manuscript received June 24, 1986. isting boundary condition. At a metallic boundary of aThe author is with the Department of Electrical Engineering and Corn- .... .ptrSince Universt of Mihian An Arbr Mi 48i09.1 microwave guide, the continuity relationships of the tan-IEEE Log Number 8610807. gential F and H fields favor the existence of only the TB
44 IEEE TRANSACTIONS ON EDUCATION, VOL. E-30, NO. 1, FEBRUARY 1987
or TM modes along the guide. No fields can exist outside of these modes can have a twofold degeneracy, raising theof the guides. In the dielectric waveguide, all six field total possible modes to 12 within 3.832 < u < 5.136.components can exist along the guide boundary, the Higher order mode groups have correspondingly largerboundary between the cylindrical core and its cladding. combination modes. It is therefore absolutely necessaryThe propagating waves which are confined to the core of to limit u, or the size of the fiber to within 7-10 icm inthe fiber sustain a variety of modes. Those with a strong order to have a single mode structure.E, field compared to the magnetic Hz field along the di- The designation of the modes from the characteristicrection of propagation are designated as the EH modes. equation is only possible after we made an approximationLikewise, those with a stronger Hz field are called the HE to the general characteristic equation. Furthermore, by as-modes. These are the hybrid modes, consisting of all six suming An (= n- n2) very small, where n1 and n2 arefield components and possess no circular symmetry. Oc- the refractive index of the core and cladding of the fiber,casionally, some circularly symmetric TE and TM modes respectively, Gloge [12] has shown, at least for the lower-can also exist. Other existing modes which are unwanted order modes, that the combination modes have the electricconsist of the radiating and the evanescent modes. The field configuration resembling a linearly polarized pat-propagating modes are found to be discrete in nature while tern; thus, he named these the linearly polarized LPimthe radiating and evanescent modes are continuous. The modes. The fundamental HE11 mode is named the LPo1propagating modes, like those that exist in metallic wave- mode, the TEO,, TMo0, and HE21 combination modes asguides, require two indexes to identify a given mode, such the LP11 mode, etc. Due to the relative simplicity of thisas HEim and EHim modes. Here 1, an integer, 0, 1, 2, - - * notation, it has been universally adopted for mode des-is the constant introduced in the analysis to separate the ignation for the fiber guides.variables in the scalar wave equation and m, the other in-teger, 1, 2, 3, * indicates the mth roots of the Bessel THE CHARACTERISTIC EQUATIONfunctions J1 and K1 which are the Bessel function of the Consider an ideal optical fiber. It consists of a longfirst kind and the modified Bessel functions, respectively. length of a uniform cylindrical core of dielectric materialThus, for each 1, there are m possible roots, m = 0, 1, 2, having a refractive index n, surrounded by an equally long* ** mmax. There is one exception. For I = 0, there exist and uniform dielectric cladding having a refractive indextwo linearly polarized sets of modes, the TEom and TMom n2. The diameter of the core is small compared to itsmodes where either the E or H field in the direction of length. The diameter of the cladding is assumed to be in-propagation becomes zero. These are modes with circular finite although its actual dimension is finite. This assump-symmetry. tion will introduce very little error in the final result butThe lowest order of mode is not TEO, or TMo1 mode, it does simplify the analysis considerably. We are, there-
which has a cutoff frequency corresponding to the first fore, dealing with a uniform dielectric waveguide imbed-zero of the Jo(u) function at u = 2.405, where u is a pa- ded in a uniform dielectric medium of infinite extent.rameter containing the fiber dimension "a" and the free From a practical point of view, light waves propagatingspace wavelength X to be defined later. The lowest mode along a good light guide undergo total reflections alongis the HE11 mode, which is the fundamental mode of a the core-cladding interface all the way. The thickness ofcylindrical light guide fiber. Its cutoff frequency is zero, the cladding actually does not matter.corresponding to the first zero of the J1 (u) function at u We shall limit our discussion to the propagating modes= 0. Between 0 < u < 2.405, there exist only the HE11 only. In a later section, radiating and evanescent modesmodes. This is the only region in which the light guide will be mentioned. The effect of mutual couplings withcan be considered as a single mode guide. However, the the propagating modes are discussed in terms of powerelectric field of the HE11 mode has two polarizations or- losses.thogonal to each other. One rotates clockwise and the Assume a step-index profile. Let the core radius be a.other counterclockwise as they propagate down the guide Then, the refractive indexlength. Only through carefully decoupling these polari-zating modes can one achieve a truly single-mode single- n = n1 for 0 < r < apolarization transmission free from intermodal interfer- andences [16], [17].The next higher order modes consist of a group of n = n2 for r > a (1)
modes designated as the TEO,, TMo0, and HE21 modes.All these modes have approximately the same cutoff fre- wherequency (u = 2.405) and almost identical propagation n, > n2constant $0l With a twofold degeneracy of the HE21mode, there exist six possible modes within 2.405 < u The actual mathematical analysis follows in the routine< 3.832. At u = 3.382, the next higher order modes set manner by solving the wave equation in cylindrical co-in. ordinate system. The direction of propagation is assumedThe next higher order modes have a cutoff frequency of to be along the z-axis.
u = 3.832. These consist ofHE12, EH11, and HE31. Each The vector fields H(r, 4, z) and E(r, X, z) are ex-
YEH: GUIDED WAVE MODES IN OPTICAL FIBERS 45
pressed as and
E, H = E(r, 0), H(r, y) exp [i(t - 3z)] (2) a=i(a) ilf 1
where co is the signal frequency of propagation and j3 is w r Kl(w) a
the propagation constant. w 1 /wr\1Following the classical routine for solving the wave -B@o - K -)I e'l for r > a,
equation, we first obtain the solution of the z-componentsof the field as (8b)
urX where Jl' and Kl are the spatial derivatives of J1 and K1,9a J respectively. Similar relations can be found with the H's.
E= AJ (u) exp [il] for 0 c r c a (3a) These are
Kl(wr) Ho= -i~~~~(a)2 Bi43 1 t(urN
and (9a)
/ur\~~~ ~ ~ ~~~~~~~~~au
wr ar) andexp[i for Pr> a (4) Ho = (-)2LB-K () IKl and
- B exp [ik/] for r> a (4b) ± Awc0n2- K (-)1 e'1 for r> a.
where A and B are arbitrary constants to be determined, (9b)and u and w are parameters defined as follows. The continuity relation requires that from (8a) and (8b)
u k- /(6) A- K 1u wK1w(w)
and ko = 2Tr/X, where X is the free space wavelength (10)corresponding to the angular frequency A. Jeand Kfarethe Bessel function of the first kind and the modified Bes- and from (9a) and (9b)sel function, respectively; l is the constant used to sepa-Ami (ne ,J (u) 9+ B- -0and the variables in solving the cylindrical wave equa- tyaKelato+ ewies(wj\ lfrB ai8b=
rate \U~~~io WOI lJ1u)w1(wIKu±w22
tions and it iS an integer having values ranging from 0 tola(0, 1, 2, 3 Imx (11)
othe alsoel thattione cndhefirst kine anewnrmlzddr Nontrivial solutions exist only if the determinant of A andquency parameter V such that ~B vanishes and one obtains, after some manipulations, the
2 1/2 2ira ~ ~ ~~~~~~2
V = (U2 + W2 ) l 2= 2T (NA) (7) characteristic equation as
X~~~~~~F V2j2 - 1 J;(u) K;(w)]where NA =(fn-s_nn2)t 12 is thenumerical aperture
usedLus2w2j u J1 (u)w(K1 (wu) I
in optics to express the ability of a system to gather light. 2n Ju 1 K;w)V, the normalized frequency parameter, will be frequently + 2 (12)referred to in later discussionsg. Wn 2 U Kt ( w) jThe transverse field components crSEo Hr and Ho can Equation (12) is the exact characteristic equation for the
be expressed in terms of Ez and Hz by using Maxwell s cylindrical fiber under consideration. Notice thait t can beequations. arranged in a form such that further simplification can be
ta2 r____ (ur\ achieved by applying the appropriate approximation to be
+ tu J A rJI()1i-a introduced.1 Ji(!~~~~~)1e ~~Theoretically, for given values of n1, n2, a, and ci, one
- Bwpll -- u2 for 0< r < a can derive from this characteristic equation much mean-a J1(u) 1 a 2ingful information about the transmission properties of the
(8a) fiber. 1) Various propagating modes that can be supported
46 IEEE TRANSACTIONS ON EDUCATION, VOL. E-30, NO. 1, FEBRUARY 1987
by the fiber can be determined and designated. 2) To- useful information of the transmission properties of thegether with (7), V2 = (U2 + w2), one can solve u or w fiber as we will develop in the following sections.as a function of V and determine the propagation constantflm for each mode of interest. As 1 and m are integers, the IDENTIFICATION OF MODESobtained values of Olm are discrete. 3) A w-f diagram can We notice that the characteristic equation carries +be drawn for each propagating mode from which other signs. These signs are used to catalog the hybrid modes.properties, such as the group velocity, the transmission The upper sign associates with the HEim modes and thedelay and the dispersion relations can be derived; and 4) lower sign with the EHim. We use double subscripts 1, mthe field components can be constructed and power trans- for mode designation because for each 1 value, there aremitted and power lost can be calculated. However, one m possible solutions due to the periodic nature of the Bes-can immediately anticipate the complication involved in sel functions. To catalog the HEim modes, we choose theusing (12). First, one expects that for each I value there J, - l m, K1 - m as functions; for EHim modes we chooseare m solutions, one for each m value. Two subscripts are J + 1, m, and Kl+ 1 m.required for each mode numbering, such as [mode]lm. For example, if 1 = 1, the characteristic equation (14)Second, the solution of (12) is so involved that it is almost for the HEim modes isimpossible to compute all the solutions that satisfy this uJ1(u)/Jo(u) = wKj(w)/Ko(w). (16)identity. Exact solution for this equation is not attempted.
We can find the cutoff frequency of these modes by lettingA USEFUL APPROXIMATION w = 0, u = V. The first zero of J1 (V) occurs at zero for
Snyder and Gloge recognized the fact that the practical m = 1. This is the HE 11 mode. It has no cutoff frequency.fabrication methods such as MCVD or VAD give a small Similarly, for m = 2, the HE12 mode is cut off at V =change in index. Also, to keep the pulse dispersion small, 3.832, etc. For 1 = 2, the characteristic equation for thethe index difference must be very small. In practice, if n, HE2m mode becomes, using (15),is in the vicinity of 1.5, n2 must be chosen such that the Jo(u) Ko(w)difference nI - n2 is on the order of 0.001-0.02. In other U j1 (u) = w K.(w) (17)words, let An = nI - n2 be small.
If the approximation An -+ 0 or n1 - n2 is used, (12) The first zero of J0( V) is at V = 2.405 for the HE21 mode.can immediately be simplified. Notice that the left-hand- For m = 2, i.e., for the HE22 mode, the cutoff is at V =side of (12) becomes zero as V2 = a2k2 (n2 - n) 0 5.52. For 1 = 0, the circular symmetrical modes TEOm andand the right-hand side of (12) becomes identical, thus TMom exist. The cutoff frequencies of these modes cor-making respond to the zeros of JO(V) at V = 2.405, 5.52, etc.
E1 lj(u) 1 K' (w)l2 Thus, for 2.405 < V < 5.52, there exist three modes.|uJl(u)+ - K(w)| = 0. (13) The TEO,, TMO1, and the HE21 modes, all have the same
u J1 (u) wK1(w)j ~~cutoff frequency at V = 2.405.
If the Bessel function identities Table I shows the order of appearance of the modeswithin the frequency ranges appearing at the left. Since V
Jl (u) = +JIT I(u) + - J/(u), is proportional to the radius of the fiber, for a fixed fre-u quency, variation of V implies also a variation of the fiber
radius as well. The group of modes shown within a certainKl (w) = - K1(w) - K+T1 (w) range of V indicate that those modes have approximately
w the same cutoff frequency and propagation constants al-and though they may have entirely different field distribu-
21J, (u) ~~~~~~tions.J-T1(u) = 2U1 (u) _ J (u), The total number of modes shown on the right side of
u this table comes about because each hybrid mode has a
_21K1(w) twofold degeneracy with clockwise and counterclock-Ki 1(w) = + + Kl (w) wise polarizations, thus doubling the combinations and
increasing the total propagating modes. As V increases,are used, (13) reduces to the total number of modes increases rapidly.
(~ J1(u) ~') Kl (w) Using the small-index differences approximation, Sny-1u (U)A w for n1 n2 (14) der and Gloge have developed another simplified charac-J1+, ( a)J K, + (w) teristic equation for designating the linearly polarized LPim
Thus, (12) has been reduced to a simple form. The solu- The LPim modes count the group of modes appearing to-tions to their characteristic equations, (14) or (15), yield gether as a single mode. The HE11 modes appear as LP01
YEH: GUIDED WAVE MODES IN OPTICAL FIBERS 47
TABLE I dur J____ORDER OF APPEARANCE OF VARIOUS MODES -Lu + V 1. (24)
Range of V Additional Modes Propagating Modes But from (23), J1 (u) /JO ( u) = V/ u. EliminatingJ1(u) /JO (u), we have
0-2.4048 HE11 22.4048-3.83 17 TEO,, TM01, HE21 6 du [2 + V21 v3.8317-5.1356 HE12, EH 1, HE31 12 u + 2 dV. (25)5.1356-5.5201 EH21, HE41 165.5201-6.3802 TE02, TM02, HE22 20 Since u << V, we can neglect the u term against V and6.3802-7.0156 EH31, HE51 24 .7.0156-7.5883 HE13, EH12, HE32 30 obtain7.5883-8.4172 EH41, HE61 34 du dV
(26)
mode followed by the LP11 mode which contains the group Integrating yieldsof modes designated as TEOB, TMo0, and HE21. The LP02 1mode is actually the sum of the TE02 TM02 and the HE22 ln u = -V + ln Cmodes, and the LP21 mode is the sum of the HE31 and VEH11 modes, etc. The modes are so grouped because orwithin this designated mode, the field distribution can be u = C e-l/. (27)shown to be linearly polarized. This is possible since the Om
fields in the longitudinal direction have been neglected as To find C, the integration constant, we let V oo, there-the result of the small index-difference approximation. fore, C = u '. Then,
EVALUATION OF THE PROPAGATION CONSTANTS U = UX e/V (28)
The information of 3, the propagation constant of the u om is the root of J0 ( u ) for V values far from cutoff. Forpropagating modes, is contained in the parameters u and I-=0, the root of Jo(u) is 2.405.w. By solving the characteristic equation of a certain mode In general, for Im modes, in the limiting case w -+ oo,together with the equation u2 + W2 = V2, the information we find
Lof (HE(V)is oband.Freape,weslvfqutorsuth fudm2 a=V l =U [I (9and u J1 (u)/Jo(u) = w K1 (w)/Ko(w) to obtain, say, uVas a function V. We do this by two limiting approaches, where ul m is the mth root of Jj(u). Some of these valuesthe approximation near cutoff and that far from cutoff. At of u 1 for the corresponding modes arecutoff, ( = kon2 or w -* 0, u = V, and
= 7.016 TMO2, TE02, HE22 LP12, etc.w =1.122 exp VJ0(V) (21) An explicit functional relationship between u and V for
the limiting case w -- oo can be derived by taking theFor the limiting case far from cutoff, we let w - oo, w derivative of the general characteristic equation (18) and
- Vand replacing w2 by V2 - u2, where u, is the cutoff value.The following is then obtained [18].
Differentiating (23) with respect to Vand using the Bessel Equation (30) is good for all LP modes except the LP01function identities, we obtain mode. In this case Gloge [12] has found
48 IEEE TRANSACTIONS ON EDUCATION, VOL. E-30, NO. 1, FEBRUARY 1987
HE5 EH F (ur\~~~6.38 - TE02' TM02? HE22 Ji+I1
64[I'E12 Ez sin (1 + 1)0
H 12 2k0an, J1(u)4 - 3.832 2w "HE3 EHt-
31, I 1JIl() )] <a
uV405=°' m-I : TEolX TMolt HE21 + sin (1 - 1)0 for r < a, (35a)
O 2 3 4 5 6 7 8 9 10 /wr\V K +lIiE1w a1(~
Fig. 1. Parameter u as a function of V for various groups of modes [24]. Ez= _ L K() sin (1 + 1)0
u(V) = (+V)V(32)K111 + (4+ V4)1/4 - sin (1 - for r > a, (35b)
A plot of u versus V for various group of modes is shown K,(w)in Fig. 1. F (ur
FIELD DISTRIBUTION iE u
Once u and w are obtained, we shall turn our attention 2k0z0a J1(u)to calculate the field distributions in the fiber. For the ap- -ur-proximation that An = n, - n2 is small, Gloge recognized Ji- Ithat the field expressions are considerably simpler in ap- - \a/ cos (1- for r <a (36a)pearance if they are expressed in Cartesian coordinates J1 (u)rather than in cylindrical coordinates [12]. Assume thatthe transverse field components are essentially linearly wr
polarized for the LP modes, these field components can HKw + a)be expressed as z Lcos (l + 1)0
2K0z0a K1(w) cs( )
Ey El J/ (-) cos 10 for r < a (33a) wrMU1) \a/ K11I~ )] >a
and + K1(w) cos (1 -I)0 forr > a. (36b)
K()Cos 10for r > a (33b) If one wishes to find E, and Ho and use boundary condi-K(w) \a/
tions at r = a to find the characteristic equation, one findswhere El is the electric field strength at the core-cladding that the same form persists.boundary. In the following, we shall investigate the field distri-The choice of using cos 10 instead of sin 10 is entirely butions of several lower order modes.
arbitrary. Again, using Maxwell's equations, we obtain 1) The lowest mode is designated as the HE,1 or LPo0the following equations by letting f = k0n, = k0n2 mode, which is the fundamental mode. It is the dominant
mode within the range- n,Ey forr< a (34a) 0< Vc 2.405.
Hx= The fields of the HE,1 mode, from (33) are given by_ 2 Ey for r > a (34b) E =ZH = EoX
yO Ey x,y =E
where (
°0ko 1 Jo(u) O.c r <Ja, (37a)
z0 is the plane-wave impedance in a vacuum.In this approximation, the Ex and Hy components are K0tr
very small compared to Ey and Hx.| ° aThe z-components of the field can be obtained by ap- K0(w) ,a-.a, (37b)
plying Maxwell's equation again.
YEH: GUIDED WAVE MODES IN OPTICAL FIBERS 49
{fta - / _-* X
E HE TE0 TM0
t -0-t LP (
t ~~01(HE ) HE21 HE21 HE12
Fig.ndmagnetic field distributions (E. and H) for an LPO Fig. 3. The electric field vector of the component modes, the TEO,, TM01,Fig.H2.EElectric,anmagneticfildedistributions(Eanand the two polarizations of the HE21 modes of the LP,, mode.(HE,,) mode.
Ez= (sin , cos )Kan2
(uJI(urla) IJo(u), O r <a, (38a) TE0+HE21 TEo-HE21 TM0 +HE TM -HE
wKi (wrla) IKO (w), r > a, (38b Fig. 4. The electric field vector of four independent linear contributionstwK1 (wr/a)/Ko(w), r > a, (38b) of the TEO1, TMO1, and HE21 modes. (TEO1 + HE21, TEOI - HE21, TMO1
+ HE21, and TMo1 - HE21).and
Hz = (cos q, sin ) andKan2z2 + w2 =V2.
uJl (ur/a)/JO(u), 0 c r < a, (39a) The amplitude El can again be calculated by normalized
wK1(wr/a)/KO(w), r > a. (39b) E 2zo 1 ul Ka(Wn)Either E., or Ey can be taken as zero, i.e., sin o is for Ez = (2) 1[K/2K(w) (42)and cos 0 is for Hz if E, = 0; and cos 1is for Ez and sin4 is for Hz for Ey = 0. zo is the vacuum impedance. Eo is The individual field distribution is sketched in Fig. 3.the amplitude which can be determined from the power The HE21 mode is a hybrid mode with a twofold degen-relations to be discussed later [23]. eracy. There are two possible field configurations as is
u KO(w) ( 2zo 1/2 w JO (u) ( 2zo 1/2 shown in this figure. All these modes have nearly the same= - 1 2 ) 2 fl-values and cutoff characteristics; they usually occurVK, (w)\ ra n2 / VJ(u )~ -ra n2 / simultaneously.
(40) There are four independent linear combinations of thesethree modes, namely the TEO, + HE21, TE01 - HE21,
The corresponding Ey and H, fields are distributed in the TM21 + HE21 and TM1 - HE21 mode. The field distri-plane of a circular cross section as is shown in Fig. 2. butions are shown in Fig. 4 with their respective com-
2) Second-order mode: When the frequency range is bined field distributions. Notice how these distributionswithin 2.405 s V s 3.832, the field structure becomes constitute the linearly polarized modes as a single linearcomplicated. As we have indicated earlier, besides the electric field vector.TEO, and TMo1 modes there exists an HE21. The set is Together with the TEO, and TMo1 modes, there are sixcalled the LP11 mode. We have different modes that can appear in the propagating fiber
J1(u1r/a) within this frequency range.Eyx = El(cos 4, sin 4)) , 0 c r c a 3) The third-order modes consist of HE12, EH11, and
J1(u1) HE31 modes which appear within the frequency range(41a) 3.832 c V c 5.52. The amplitude of the third-order
modes can be evaluated in a similar manner and their fields
Ey, = El (cos 4, sin 4) K( r/a) r - a (41b) sketched likewise. A total of 12 combinations of fields areK1(w1) possible.
where u, and w, are the value of u and w within this fre- As V increases, or for a fixed frequency, as the diameterquency range. There are four possible combinations of of the core increases, the number of propagating modesE(x, y) and E1 (cos 4), sin 4)) and u1 and W1, all satisfying increases rapidly.these equations. It is very interesting to note that if the guided wave
intensity is plotted as a function of r/a for a fixed fre-UJ2 (u1) =WK2(W1) quency (the normalized frequency V), a curve resembling
Ul J1(fu.) K1 (.{W1)4 a Gaussian curve is obtained, at least for the lower order
50 IEEE TRANSACTIONS ON EDUCATION, VOL. E-30, NO. 1, FEBRUARY 1987
I I, LPo LPI LP2 LP2 LP3 LP0l "II "02 12 0O3 LP131.0 0
1.0 1.0
V 2.4 V = .8 0.8 -0.2
0.5 05- 06 0.4
ci'0.4 --0.6 a-L'O 2 3 r/a 0 2 3 r/a
Fig. 5. Shape of the guided wave intensity (I(r)) as a function of the 02 0.8radius of the fiber for several values of V. I(r) is the power density ofthe guiding mode for constant total guiding power. 0 2 4 6 8 10
V
modes. Fig. 5 shows such a plot. Therefore, it is some- Fig. 6. Plot of the normalized optical power in the case and cladding as a
times useful to approximate the field distribution of the function of V for lower order LP modes.
LPO mode by an exponential expression and make thecalculation of the power relations much simpler. in the cladding. For example, for the LPOI mode, if V <
2, 80 percent of the power is lost in the cladding. To keepPOWER DISTRIBUTION 90 percent of the power in the core, V must be chosen
The total power carried by a particular fiber mode along about five. This could set a limit to how small the corethe z-direction is given by the following integral [14]. radius can be designed for single-mode fiber. For the LPoI
X 2r mode, about 30-50 percent of the fundamental power is
Pt= , 3 Re [E x H*] - 1,rdr do (43) carried by the cladding.20 Further power loss in fibers can be introduced by fiber
where Re indicates the real part, x indicates the vector irregularities. In fact, any dielectric waveguide will ra-prduct denoesthescaar u , * idiate if it is not absolutely straight. For a bend, for ex-product, * denotes the scalar product, * ndicates the y
complex conjugate and I z is a unity vector in the z direc- ample, the radius of curvature of the bend can affect thetion. Using the values of E and H for the particular mode radiation loss [19]-[21], [23]. For the radius of curvaturefrom (33)-(36), one can integrate (43) to obtain the total R > a, fundamental power is lost through coupling topower, including the power transmitted into the core and higher order modes and/or radiation modes.the power lost in the cladding. The amplitude form El in(33)-(36) is evaluated by normalizing Pt to unity at the CONCLUSIONcore-cladding interface for that mode. For example, (40) We have presented a logical and simplified descriptionfor Eo and (42) for El are the result of the normalization of the mode structure of a cylindrical optical fiber. Lin-process. early polarized mode designation is emphasized. Exam-The fractional power carried by the fiber core and that ples of some lower order modes are illustrated in mode
carried by the cladding can be obtained by changing the designations, calculation of the propagation constants, thelimit of the first integral of (43) from I' to I' and 1', u-V diagram, the field configurations, and the power re-respectively, and then to divide it by Pt. For the LPOI lations. Formulas are introduced when necessary withoutmode, we obtain lengthy derivations as these can be found elsewhere in the
p_ (a \ 2 Ko (w 21 references. This helps to concentrate our attention on thePcore ) L YK1(w) j (44) physical description of the problem.
and REFERENCES___ (u\ 2 s. toward
clad = (o j(45) optical-fiber transmission systems," Proc. IEEE, vol. 61, pp. 1703-
Pt V ~KK,)- (w)/ -' 1752, Dec. 1973. A comprehensive list of 260 papers and books isincluded.
Adding (44) to (45) resulted in Pt = Pcr + pcld as is [2] D. Botez and G. J. Herskowitz, "Components for optical communi-= Pcoe cladas iscation systems," Proc. IEEE, vol. 68, pp. 689-732, June 1980. A
expected. list of 640 papers is included.
Pclad * zero indicates that that amount of power is lost. [3] Special Issue on Optical-Fiber Communications, Proc. IEEE, vol.
Power is lost either through radiation or as heat loss 68, pp. 1172-1327, Oct. 1980.[4] Y. Suematsu, "Long-wavelength optical fiber communication," Proc.
through evanescent modes. Fig. 6 is a plot of Pc1ad /IP, and IEEE, vol. 71, pp. 692-721, June 1983. A list of 461 articles, mostlyscore /Pt as a function of V for various lower order modes. published after 1980 is included.It is noticed that for larger values of V, more and more [5] J- A. Stratton, Electromagnetic Theory. New York: McGraow-Hill,power will be carried onto the core. As V = (2ira / X\) [6] E. Snitzer, "Cylindrical dielectric waveguide modes," J. Opt. Soc.(NA), larger V means the fiber is gathering more light. Amer., vol. 51, pp. 491-498, May 1961.This happens only in mnultimode fibers as a / ) is large. [7] G- Biernson and D. J. Kinsley, "Generalized plots of mode patterns
*or * as r r 11 XT ~~~~~~~~~ina cylindrical dielectric waveguide applied to retinal cones,"' IEEEFor simple-mode fibers, V iS usually best Kept small. lNo- Trans. Microwave Theory Tech., vol. MTT-13, pp. 345-356, Maytice that below cutoff values of V, all power is consumed 1965.
YEH: GUIDED WAVE MODES IN OPITICAL FIBERS 51
[8] R. D. Mauren, "Glass fibers for optical communications," Proc. [23] L. B. Jeunhomme, Single-Mode Fiber Optics. New York: MarcelIEEE, vol. 61, pp. 452-462, Apr. 1973. Dekker, 1983.
[9] D. Marcuse, Theory of Dielectric Optical Waveguide. New York: [24] A. W. Snyder and J. D. Love, Optical Waveguide Theory. London,Academic, 1974. England: Chapman and Hall, 1983.
[10] T. Li, "Structure, parameters, and transmission properties of opticalfibers," Proc. IEEE, vol. 68, pp. 1175-1180, Oct. 1980.
[11] A. N. Snyder, "Asypmtotic expression for eigenfunctions and eigen-values of a dielectric or optical waveguide," IEEE Trans. MicrowaveTheory Tech., vol. MTT-17, pp. 1130-1138, Dec. 1969.
[12] D. Gloge, "Weakly guided fibers," Appl. Opt., vol. 10, p. 2252,Oct. 1971.
[13] A. W. Snyder and D. J. Mitchell, "Leaky rays on circular optical Chal Yeh (A'48-SM'51-LS'81) received thefibers," Opt. Soc. Amer., vol. 64, pp. 599-607, May 1974. B.S.E.E. degree from Zhejiang University,
[14] R. Olshansky, "Propagation in glass optical waveguides," Rev. Mod. Hangzhou, China, in 1931, the Master degree inPhys., vol. 51, pp. 341-367, Apr. 1979. electrical engineering, and the Ph.D. degree in
[15] S. A. Schelkunoff, Electromagnetic Waves. Princeton, NJ: Van applied physics, both from Harvard University,Nostrand, 1943. Cambridge, MA, in 1934 and 1936, respectively.
[16] L. Eyges, P. Gianiuo, and P. Wintersteiner, "Modes of dielectric He joined Peiyang University in Tientsin,waveguides of arbitrary cross sectional shape," J. Opt. Soc. Amer., China as a Professor of Electrical Engineering invol. 69, pp. 1226-1235, Sept. 1979. 1936. In 1937 he joined the Tsinghau University
[17] R. B. Dyott, J. R. Cozens, and D. B. Morris, "Preservation of po- in Peking, China, as a Professor of Electrical En-larization in optical-fiber waveguides with elliptical cores," Electron. gineering. During the Sino-Japanese war (1937-Lett., vol. 15, pp. 380-382, June 1979. 1945), Tsinghau University was moved to Kumming and merged with the
[18] P. K. Cheo, Fiber Optics. Englewood Cliffs, NJ: Prentice-Hall, Nankai and Peking Universities to form the Associated South-Western Uni-1985. versity, At the end of the War, Tsinghau University was moved back to
[19] D. Marcuse, "Microbending losses of single-mode, step-index and Peking in 1946. He continued as Chairman of the Department of Electricalmultimode, parabolic-index fibers,'" Bell Syst. Tech. J., vol. 55, pp. Engineering until 1947 when he was granted a sabbatical leave to visit937-955, Sept. 1976. Harvard University. From 1948 to 1956 he was a Visiting Professor at the
[20] K. Petermann, "Theory of microbending loss in monomode fibers University of Kansas, Lawrence. He joined the University of Michigan,with arbitrary refractive index profile," Acta Electron. Ubertragung, Ann Arbor, first as a Research Engineer at Willow Run Laboratories andvol. 30, pp. 337-342, Sept. 1976. as a Lecturer in the Department of Electrical Engineering of the University.
[21] J. Sakai and T. Kiumura, "Practical microbending loss formula for In 1961 he was transferred to the Department of Electrical Engineering assingle-mode optical fibers," IEEE J. Quantum Electron., vol. QE- an Associate Professor and promoted to full Professor in 1964. He re-15, pp. 497-500, June 1979. mained at the University of Michigan until his retirement and became a
[22] J. E. Midwinter, Optical Fiberfor Transmission. New York: Wiley, Professor Emeritus of the Department of Electrical Engineering and Com-1979. puter Science in 1981.
