Zhang K., Li D. Electromagnetic Theory for Microwaves and Optoelectronics

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6.6 Circular Dielectric Waveguides and Optical Fibers 371

Figure 6.22: Dispersion curves for some lower modes in a circular dielectric

waveguide (a) and weakly guiding ﬁber (b).

(b) n > 1: For n > 1, the curves are similar to those for n = 1. As an

example, the curves for n = 2 are shown in Fig. 6.21(b).

The right-hand side of (6.259) for EH

modes is a monotonically de-

creasing function of T a which has value zero at T a = 0 and approaches

−∞ as T a gets near V and τa =

− (T a)

goes to zero. The left-

hand side of (6.259) is a multi-branched function of T a which behaves like

tan(T a −π/4) for n even or −cot(T a −π/4) for n odd when the value of T a

is large enough. A representative plot of (6.259) for n = 2 is shown in the

lower half of Fig. 6.21(b).

The right-hand side of (6.260) for HE

modes is a monotonically in-

creasing function of T a which has value zero at T a = 0 and takes the value

(V ²

)/[2(n − 1)²] as T a approaches V and τa =

− (T a)

approaches

zero. The left-hand side of (6.260) is a multi-branched function of T a, which

behaves like −tan(T a −π/4) for n even or cot(T a −π/4) for n odd when the

value of T a is large enough. The function has no root when T a is less than

the value for the function equals (T a)/[2(n − 1)]. A representative plot of

(6.260) for n = 2 is shown in the upper half of Fig. 6.21(b). The dotted line

with slope 1/2(n − 1) represents the lower possible value of T a for the left-

hand side of (6.260) having root, and the dotted line with slope ²

/2(n −1)²

372 6. Dielectric Waveguides and Resonators

represents the value of the right-hand side of (6.260) with T a = V .

In Fig. 6.21(a), (b) and the similar ﬁgures for n > 2, the intersections

of the lines plotted by functions on left-hand and right-hand sides of the

equations (6.259) or (6.260) represent the guided modes in the waveguide.

The values of T , τ, and β versus ω are then calculated and the dispersion

relations are obtained. The dispersion curves for some lower modes in circular

dielectric waveguide are shown in Figure 6.22a.

(3) Cutoﬀ Conditions for Circularly Asymmetric Modes

The solutions of the eigenvalue equations (6.259) and (6.260) under the con-

dition of T a = V and τ = 0 give the cutoﬀ conditions of the waveguide. To

do this, we have to examine equations (6.259) and (6.260) for small values of

τa, using the following asymptotic expressions of modiﬁed Bessel functions

for small arguments derived from (C.9),

lim

x→0

(x)

= lim

x→0

−

n−1

(x)

=−

−

2(n − 1)

, for n > 1, (6.262)

lim

x→0

(x)

= lim

x→0

−

(x)

= −

−ln

γx

, for n = 1, (6.263)

where γ is Euler’s constant, γ = 1.781.

With these small-value approximations, the eigenvalue equation of EH

modes (6.259) for small τa reduces to

n+1

(T a)

= −

2n² T a

τa

. (6.264)

Hence the cutoﬀ conditions for EH

modes (τa = 0, T a = V ) are given by

(T a) = 0, T

a = x

= V, ω

√

− µ

, (6.265)

where x

denotes the mth root of J

(x) = 0.

In the same approximation, the eigenvalue equation for HE modes (6.260)

for small τ a reduces to

n−1

(T a)

T a

γτa

, for n = 1, (6.266)

n−1

(T a)

T a

2(n − 1)²

, for n > 1. (6.267)

The cutoﬀ conditions for HE

(n = 1) modes (τa = 0, T a = V ) (6.266)

are therefore

(T a) = 0, T

a = x

= V, ω

√

− µ

, for n = 1. (6.268)

6.6 Circular Dielectric Waveguides and Optical Fibers 373

Note that, T a = 0 is the ﬁrst root of equation (6.268), denoted by HE

mode.

For HE

(n > 1) modes, by substitution of the recursion relation

n−1

(x) =

2(n − 1)

n−2

(x) + J

(x)],

the above equation for n > 1 (6.267) is equivalent to

n−2

(T a)

= −

∆²

, for n > 1, (6.269)

which gives

a = z

= V, ω

√

− µ

, for n > 1, (6.270)

where z

is the mth root of

n−2

(z)

= −

∆²

− ²

+ ²

The vertical asymptotes of the dashed lines in Fig. 6.21(a) and the lower

half of Fig. 6.21(b) represent the solutions of (6.265) and (6.266). The dotted

line with slope ²

/2(n − 1)² in the upper half of Fig. 6.21(b) represents the

right-hand side of (6.267). Its intersections with the multi-branched curves

representing the function J

n−1

(T a)/J

(T a) identify the minimum values of

T a for which the corresponding ﬁelds can be completely conﬁned in the

waveguide without transverse radiation. These are just the cutoﬀ conditions

for those modes (6.270).

We showed in Section 6.6.1 that, T a = 0 is not the cutoﬀ condition for

TE and TM modes. Similarly, we can see that, the eigenvalue equation of

modes and that of HE

(n > 1) modes for small τa, (6.264) and

(6.267), can not be satisﬁed by τ a → 0 and T a → 0. So T a = 0 is not the

cutoﬀ condition for all EH

modes and the HE

modes with n > 1. Only

the eigenvalue equation of HE

(n = 1) modes for small τa (6.266) can be

satisﬁed by τa → 0 and T a → 0. So T a = 0, i.e., T

= 0 is the ﬁrst cutoﬀ

condition of HE

modes, labeled by m = 1, i.e., HE

mode. This is just

the lowest mode, i.e., dominant mode for circular dielectric waveguide with

zero cutoﬀ frequency.

In summary, we have the cutoﬀ conditions of circular dielectric waveguide

for all the following eigenmodes.

(1) If n = 0, the cutoﬀ conditions for both EH

and HE

modes are

a) = 0, T

, (6.271)

where x

is the mth root of the zeroth order Bessel function. The EH

modes are labeled TE

modes and the HE

modes are labeled TM

modes.

374 6. Dielectric Waveguides and Resonators

(2) The cutoﬀ conditions for EH

modes are

a) = 0, T

n,m

, (6.272)

(3) If n = 1, the cutoﬀ conditions for HE

modes are

a = 0, T

= 0, (6.273)

and

a) = 0, T

1,m−1

, (6.274)

where T

= 0 is recognized as the ﬁrst root, m=1, and x

1,m−1

is the (m−1)th

root of the ﬁrst order Bessel function, m = 2, 3, 4, ···.

(4) If n > 1, the cutoﬀ conditions for HE

modes are

n−2

(T a)

− ²

+ ²

, T

n,m

, (6.275)

where z

are the roots of the above equation, and m is the number labeling

of the roots.

(4) Behavior of EH and HE Modes

Let us now examine the behavior of the EH, HE, TE and TM modes.

Rewrite the deﬁnition of parameter χ (6.214) for nonmagnetic dielectric

waveguide

χ=

(T a)

(τa)

(T a)

T aJ

(T a)

(τa)

τaK

(τa)

, (6.276)

χ=

(T a)

T aJ

(T a)

(τa)

τaK

(τa)

(T a)

(τa)

, (6.277)

and

jβχ

ωµ

. (6.278)

From (6.232) and (6.233), for µ

= µ

, we have the eigenvalue

equations in the following two equivalent forms

(T a)

T aJ

(T a)

(τa)

τaK

(τa)

∆²K

(τa)

τaK

(τa)

(

∆²K

(τa)

τaK

(τa)

+ n

(T a)

(τa)

¸·

(T a)

(τa)

¸¾

1/2

, (6.279)

6.6 Circular Dielectric Waveguides and Optical Fibers 375

(T a)

T aJ

(T a)

(τa)

τaK

(τa)

= −

∆²K

(τa)

τaK

(τa)

(

∆²K

(τa)

τaK

(τa)

+ n

(T a)

(τa)

¸·

(T a)

(τa)

¸¾

1/2

, (6.280)

where ∆² = (²

− ²

)/2. It is to be noted that, although (6.279) and (6.280)

are invariant when n is replaced by −n, but according to (6.276) and (6.277),

χ changes sign under this substitution, and the ﬁelds are therefore diﬀerent

for n and −n.

By the convention adopted in the literature, we label the modes belonging

to (6.232) or (6.259) or with the plus sign on the radical in (6.279) and (6.280)

as EH modes, and the modes belonging to (6.233) or (6.260) or with the minus

sign on the radical in (6.279) and (6.280) as HE modes.

Substituting (6.280) into (6.277), we ﬁnd that, for EH mo des with a plus

sign on the radical in (6.279) or (6.280), |χ| > 1, i.e.,

χ ≥ 1, for EH

modes, n > 0,

χ ≤ −1, for EH

modes, n < 0,

χ → ∞, for EH

or TE

modes, n = 0.

Substituting (6.279) into (6.276), we ﬁnd that, for HE modes with a minus

sign on the radical in (6.279) or (6.280), |χ| < 1, i.e.,

−1 < χ < 0, for HE

modes, n > 0,

0 < χ < 1, for HE

modes, n < 0,

χ = 0, for HE

or TM

modes, n = 0.

These inequalities may be proven by estimation with the realization that τa

is a pure real quantity and K

(τa)/τaK

(τa) is accordingly an intrinsically

negative, real quantity.

From (6.278), we ﬁnd that, the phase of H

leads that of E

by π/2 for

the wave with χ > 0, and the phase of H

lags that of E

by π/2 for the wave

with χ < 0. From the above description of χ and n for diﬀerent modes, we

see that, when n > 0, i.e. for counterclockwise skew waves, for EH modes,

χ ≥ 1 > 0, the phase of H

leads that of E

by π/2 and for HE modes, χ < 0,

the phase of H

lags that of E

by π/2; and when n < 0, i.e., for clockwise

skew waves, for EH modes, χ ≤ −1 < 0, the phase of H

lags that of E

π/2 and for HE modes, χ > 0, the phase of H

leads that of E

by π/2.

From the ﬁeld-component expressions (6.218)–(6.229), we notice that the

transverse ﬁeld component is an elliptically polarized ﬁeld composed of two

circularly polarized ﬁelds in opposite senses. First we consider a clockwise

skew wave, n < 0. For EH modes, χ is negative and the sense of the domi-

nant circularly polarized wave components of E and H are counterclockwise

376 6. Dielectric Waveguides and Resonators

Figure 6.23: Behavior of EH and HE Modes.

circularly polarized, so that the composed E and H are counterclockwise

elliptically polarized. This describes the elliptically polarized wave has trans-

verse ﬁeld vectors that rotate in a sense opposite to that of the advancing

skew wave. For HE modes, on the other hand, χ is positive and the dominant

circularly polarized transverse components of E and H are clockwise, so that

the elliptically polarized wave whose transverse ﬁeld vectors have the same

sense of rotation as the advancing skew wave. Second we consider a counter-

clockwise skew wave, n > 0. For EH modes, χ is positive and the transverse

ﬁelds are clockwise elliptically p olarized whereas for HE modes, χ is negative

and the transverse ﬁelds are counterclockwise elliptically polarized.

Finally we conclude that the transverse ﬁeld vectors in the elliptically

polarized skew wave of EH modes rotate in an opp osite sense as the skew

wave, refer to Fig. 6.18, whereas the transverse ﬁeld vectors of HE modes

rotate in the same sense to the skew wave. The above relationship is shown

in Fig. 6.23.

Recall that, as we have mentioned before, in this book, the direction

of the rotation of ﬁeld vectors is determined by an observer who transmits

the wave, i.e., looks in the direction of propagation. In some literature, the

direction of the rotation of ﬁeld vectors is determined by an observer who

receives the wave, i.e., looks in the opposite direction, and the clockwise and

counterclockwise of the rotation of polarized ﬁeld vectors are exchanged.

As we mentioned before, for waveguides made by isotropic material and

isotropic boundaries, the longitudinal phase coeﬃcients for clockwise skew

wave, n < 0, and for counterclockwise skew wave, n > 0, are the same.

Under this condition, the result of superposition of two opposite skew waves

6.6 Circular Dielectric Waveguides and Optical Fibers 377

is a wave of ﬁelds with sin nφ or cos nφ dependence, i.e., even symmetric or

odd symmetric modes. These are two orthogonal degenerate mo des.

For n = 0, the EH

and HE

modes become TE

and TM

modes.

They become circularly symmetric modes, i.e., meridional waves.

The ﬁeld patterns of some lower modes in a circular dielectric waveguide

are shown in Fig. 6.20.

6.6.3 Weakly Guiding Optical Fibers

In typical optical ﬁbers, the refractive index of the core is only slightly larger

than that of the cladding. The diﬀerence between the refractive indices of

the core and the cladding is in the range of 1%–5%. So that (n

−n

)/n

¿ 1

and ²

≈ ²

. This kind of optical ﬁber is known as the weakly guiding optical

ﬁber.

For weakly guiding optical ﬁber, the critical angle on the boundary is

rather large. For guided modes, the angle of incidence must be larger than

the critical angle and close to π/2, i.e. the wave vector of the incident wave

is almost parallel to the z axis. In this case, the longitudinal components of

the ﬁelds are much less than the transverse components, and the longitudinal

wave number is approximately equal to the wave number of a plane wave in

the core material, T ¿ β, β =

− T

≈ k

; that is to say, the wave is

close to the TEM mode.

(1) Eigenvalue Equations for Weakly Guiding Fibers

For weakly guiding optical ﬁbers, ²

≈ ²

and µ

= µ

. The eigenvalue

equation (6.258) becomes

n+1

(T a)

T aJ

(T a)

n+1

(τa)

τaK

(τa)

¸·

n−1

(T a)

T aJ

(T a)

−

n−1

(τa)

τaK

(τa)

= 0. (6.281)

It can be separated into two equations:

T aJ

(T a)

n+1

(T a)

= −

τaK

(τa)

n+1

(τa)

, for EH modes, (6.282)

T aJ

(T a)

n−1

(T a)

τaK

(τa)

n−1

(τa)

, for HE modes. (6.283)

These two eigenvalue equations can also be obtained by applying the weakly

guiding approximation ²

≈ ²

to equations (6.259) and (6.260) .

From the recurrence formulas for Bessel functions (C.13) and (C.15), we

have

(x) =

2(n − 1)

n−1

(x)−J

n−2

(x), K

(x) =

2(n − 1)

n−1

(x)+K

n−2

(x).

378 6. Dielectric Waveguides and Resonators

Applying them in (6.283), yields,

T a

2(n−1)

T a

n−1

(T a) − J

n−2

(T a)

n−1

(T a)

τa

2(n−1)

τa

n−1

(τa) −K

n−2

(τa)

n−1

(τa)

After rearrangement, the eigenvalue equation of HE modes for a weakly guid-

ing ﬁber becomes

T aJ

n−2

(T a)

n−1

(T a)

τaK

n−2

(τa)

n−1

(τa)

, for HE modes. (6.284)

Under the weakly guiding condition, the eigenvalue equations for TE and TM

modes, (6.235) and (6.236), reduce to the same equation:

T aJ

(T a)

= −

τaK

(τa)

, for TE and TM modes. (6.285)

From the above discussions we see that, for weakly guiding ﬁbers, the eigen-

value equations for all of EH, HE, TE, and TM modes, (6.282), (6.284) and

(6.285) can be expressed by the universal eigenvalue equation

T aJ

−1

(T a)

= −

τaK

−1

(τa)

, (6.286)

where







1, for TE and TM modes,

n + 1, for EH modes,

n − 1, for HE modes.

(6.287)

(2) Cutoﬀ Conditions for Weakly Guiding Fibers

For weakly guiding ﬁbers, ∆² ≈ 0, the cutoﬀ conditions for HE modes (6.275)

becomes

n−2

a) = 0. (6.288)

The cutoﬀ conditions of the other modes remain valid. It can be easily seen

that, for weakly guiding ﬁbers, the cutoﬀ conditions for all EH, HE, TE, and

TM modes (6.272), (6.288) and (6.271) can be expressed by the universal

equation

−1

a) = 0, (6.289)

where n

is identical to that in (6.287).

In the special case of n = 1 for HE modes, there is a zero cutoﬀ HE

mode with the cutoﬀ condition

a = 0.

6.6 Circular Dielectric Waveguides and Optical Fibers 379

(3) Modes and Degeneracy for Weakly Guiding Fibers

The modes and degeneration for weakly guiding ﬁbers are as follows. See

Fig. 6.22b.

(1) The dominant (lowest) mode, HE

mode with zero cutoﬀ frequency is

not a circularly symmetric mode, hence it is regarded as two orthogonal po-

larized modes with the same dispersion characteristics that constitute double

degenerate modes.

(2) The cutoﬀ conditions for TE

and TM

modes are the same, and

for a weakly guiding waveguide, the eigenvalue equations are also approx-

imately the same. In this case, they are nearly degenerate modes in the

weakly guiding approximation. Furthermore, the cutoﬀ conditions and the

eigenvalue equations for the two orthogonally polarized HE

modes are the

same as those for TE

and TM

modes, so those modes possess four-mode

degeneracy in the weakly guiding approximation.

(3) When n > 1, it may be easily seen from the universal eigenvalue

equation (6.286) and the universal cutoﬀ equation (6.289) that in the weakly

guiding approximation, the EH

n−1,m

mode and HE

n+1,m

mode are degen-

erate modes, so these modes also possess four-mode degeneracy including

polarization degeneracy.

(4) When n = 1, the cutoﬀ condition for HE

1,m+1

modes are the same

as those for EH

or HE

modes. However, the eigenvalue equations are

diﬀerent for diﬀerent modes, none of these modes can be degenerate modes.

The cutoﬀ conditions for some modes in a circular dielectric waveguide

are listed in Table 6.1.

Table 6.1 Modes in a circular dielectric waveguide

number total

modes

cutoﬀ T

a of numb er

condition modes of modes

T a = 0 0 2 2

,TM

,HE

(T a)

= 0 2.405 4 6

,EH

,HE

(T a)

= 0 3.832 6 12

,HE

(T a)

= 0 5.136 4 16

,TM

,HE

(T a)

= 0 5.520 4 20

,HE

(T a)

= 0 6.38 4 24

,EH

,HE

(T a)

= 0 7.01 6 30

,HE

(T a)

= 0 7.58 4 34

,HE

(T a)

= 0 8.41 4 38

,TM

,HE

(T a)

= 0 8.65 4 42

,HE

(T a)

= 0 8.71 4 46

,HE

(T a)

= 0 9.76 4 50

,HE

(T a)

= 0 9.93 4 54

The number of modes includes the number of polarization degeneracy

In a multi-mode ﬁber, there are usually dozens, even hundreds, of guided

modes, but in a single-mode ﬁber there is only one guided mode, i.e., the

mode.

380 6. Dielectric Waveguides and Resonators

6.6.4 Linearly Polarized Modes in Weakly Guiding

Fibers

In weakly guiding optical ﬁbers, the EH

n−1,m

mode and the HE

n+1,m

mode

are nearly degenerate mo des. The two waves are elliptically polarized in

opposite senses with almost equal phase coeﬃcients. This property ensures

that the two modes with the same amplitude will add up to represent a

linearly polarized mode denoted by LP

. The state of linear polarization

will remain during the propagation because the phase coeﬃcients of the two

elliptically polarized waves in opposite senses are approximately equal.

As we have mentioned before, in the weakly guiding approximation, the

longitudinal components of the ﬁeld are negligible compared with the trans-

verse components and the longitudinal wave number is close to the space

wave number in the core material, β ≈ k

For a linearly polarized mode in an arbitrary direction on the transverse

section. We choose our coordinate system such that one of the coordinate

axis is along the electric ﬁeld vector, for an example, one deﬁnes the mode

with its transverse electric ﬁeld in the y direction as y linear polarized mode

or TE

(x)

mode. For TE

(x)

modes, U

(x)

= 0, V

(x)

6= 0, and function V can be

written as the product of a function in ρ, a function in φ, and a function in

z. To satisfy the boundary conditions on the circular cylindrical boundary,

the function must be expressed in terms of cylindrical harmonics.

In region 1, the core, to avoid singularity on the axis ρ = 0, the functions

in ρ must be Bessel functions of the ﬁrst kind. The functions in φ can either

be even functions or odd functions. So the function V

(x)

is given by

(x)

=AJ

(T ρ)

cos nφ

sin nφ

−jβz

. (6.290)

The transformation from polar coordinates to the cartesian coordinates

in two dimensions is given by

∂

∂x

= cos φ

∂

∂ρ

−

sin φ

∂

∂φ

∂

∂y

= sin φ

∂

∂ρ

cos φ

∂

∂φ

Substituting (6.290) into the ﬁeld-component expressions in terms of U

(x)

and V

(x)

, (4.153)–(4.158), and using the recurrence formula (C.13) and diﬀer-

ential formula (C.16), we obtain the ﬁeld components of the linearly polarized

modes in the core:

y 1

= −jωµ

∂V

(x)

∂z

= −ωµ

βAJ

(T ρ)

cos nφ

sin nφ

−jβz

, (6.291)

= jωµ

∂V

(x)

∂y

= −j

ωµ

n+1

(T ρ)

sin (n + 1)φ

−cos (n + 1)φ