Chapter 3 Propagation of optical beams in fibers

Introduction

Since the realization of low-loss fibers in 1970s, fibers become the most important medium for optics communication.

The technology of optical communication in fibers can be stated as that of feeding( 饲养 , 吃 , 输送 ) optical pulses at a maximal rate into one end of a fiber and retrieving( 恢复 ) them at the other end. The main goal of a communication system is to receive the pulses at the output end with minimal loss of energy, minimal spread, and minimal contamination( 玷污 , 污染 ) by noise. For example, the silica([ 化 ] 硅石 )glass fiber ,which has the character of low-loss propagation of confined optical modes, has become the most important transmission medium for long distance.

In this chapter, we will study : 1. the subject of optical guided modes in fibers

2. The problem of pulse spreading due to group velocity dispersion and various strategies of combating it

3.1 wave equation in cylindrical 3.1 wave equation in cylindrical coordinates(coordinates( 柱坐标柱坐标 ))

In Chaoter 2 we have shown that optical waveguides with a quaIn Chaoter 2 we have shown that optical waveguides with a quadratic index profile (See Equation 2.9-1a) can support guided nodratic index profile (See Equation 2.9-1a) can support guided nondiffracting modes. The effect of diffraction spreading is counterndiffracting modes. The effect of diffraction spreading is counterbalanced by the lensing effort of the index profile of the guide. Cbalanced by the lensing effort of the index profile of the guide. Commercialommercial( 商用的 ) silica-based optical fiber use a step index profile with a “high” index core and a “low” index cladding( 覆层 ). These fibers form the backbone( 脊椎 , 骨干 )of most modern communication systems, and the study of their modes of propagation is the subject at hand.

, , ,r z r zE E E H and H

Since the refractive( 折射的 ) index profiles n(r) of most fibers are cylindrically symmetric ( 柱对称性的 ), it is convenient to use the cylindrical coordinate system. The field components are , the wave equation 2.4-3 assumes its simple form only for the Cartesian( 笛卡儿 ) components of the field vectors( 矢量 ) 。

n(r)

n1

n2

2a

2b

Figure 3-1 Structure and index profile of a step-index circular waveguide

Since the unit vector ar and aφ are not constant vectors, the wave equation involving the transverse( 横向的 ) components are very complicated. The wave equation for the z component of the field vectors, however, remains simple

2222 tcnk 2222 tcnk

2 2 0Z

Z

Ek

H

3.1-1

, , ,r z r zE E E H and H

The problems of wave propagation in a cylindrical structure are usually approached by solving for Ez and Hz first and then expressing in terms of Ez and Hz

where and is the laplacian operator given by:22

2

2

2

2

22

22 11

zrrrr

2

2

2

2

22

22 11

zrrrr

Since we are concerned with the propagation along the waveguide, we assume:

( , ) ( , )exp[ ( )]

( , ) ( , )

t ri t z

t r

E r E

H r H

exp[ ( )]i t z every component of the field vector assume the same z-and t-dependence of .

3.1-2

Maxwell’s curl equations are now written in terms of the cylindrical components and are given by :

3.1-3

3.1-4

From above equation, we get

3.1-5

3.1-6

From the results, we know that these relations show it is sufficient to determine Ez and Hz in order to specify uniquely the wave solution. The remaining components can be calculated from (3.1-5) and (3.1-6)

With the assumed z-dependence of (3.1-2), thWith the assumed z-dependence of (3.1-2), the wave equation (3.1-1) becomes e wave equation (3.1-1) becomes

This equation is separable, and the solution tThis equation is separable, and the solution takes the formakes the form

where l=0,1,2,3….., so that Ewhere l=0,1,2,3….., so that El l and Hand Hz z are singlare single-valued functions of φ .Thene-valued functions of φ .Then

)71.3(01 22

2

2

2

2

z

z

H

Ek

rrr )71.3(0

1 222

2

2

2

z

z

H

Ek

rrr

)81.3)(exp()(

ilr

H

E

z

z )81.3)(exp()(

ilr

H

E

z

z

(3.1-7) becomes (3.1-7) becomes where ψ=Ewhere ψ=Ez z and Hand Hz.z.

Equation (3.1-9) is the Bessel differential equatEquation (3.1-9) is the Bessel differential equation, and the solutions are called Bessel functioion, and the solutions are called Bessel functions of order l. If kns of order l. If k22-β-β22>0, the general solution of >0, the general solution of (3.1-9) is (3.1-9) is

01

2

222

2

2

r

lk

rrr0

12

222

2

2

r

lk

rrr

)()()( 1211 hrYchrJcr )()()( 1211 hrYchrJcr

3.1-9

3.1-10

where hwhere h22=k=k22-β-β22, c, c1 1 and cand c22 are constants, and J are constants, and J11,Y,Y11 are Bessel functions of the first and second kiare Bessel functions of the first and second kind, respectively, of order l. If knd, respectively, of order l. If k22-β-β22<0, the gen<0, the general solution of (3.1-9) is eral solution of (3.1-9) is

where qwhere q22=β=β22-k-k22, c, c1 1 and cand c22 are constants, and I are constants, and Ill，，KKl l are the modified Bessel functions of the firare the modified Bessel functions of the first and second kind, respectively, of order l.st and second kind, respectively, of order l.

)()()( 1211 qrKcqrIcr )()()( 1211 qrKcqrIcr 3.1-11

To proceed with our solution, we need the To proceed with our solution, we need the asymptotic forms of these functions for asymptotic forms of these functions for small and large arguments. Only small and large arguments. Only leading terms will be given from leading terms will be given from simplicity.simplicity.

For x<<1:For x<<1:

asymptotic [ 数 ] 渐近线的 , 渐近的

....5772.02

ln2

)(

2

1)(

xxY

x

nxJ

o

t

t

....5772.0

2ln

2)(

2

1)(

xxY

x

nxJ

o

t

t

In these formulas l is assumed to be a nonnegaIn these formulas l is assumed to be a nonnegative integer. The transition from tive integer. The transition from

the small x behavior to the large x asymptotic fthe small x behavior to the large x asymptotic form occurs in the region of x~l.orm occurs in the region of x~l.

For x>>1,l:For x>>1,l:

42cos

2)(

2/1

lx

xxJ l

42cos

2)(

2/1

lx

xxJ l

42sin

2)(

2/1

lx

xxYl

42sin

2)(

2/1

lx

xxYl

xl e

xxI

2/1

2

1)(

x

l ex

xI2/1

2

1)(

ex

xK l

2/1

2)(

ex

xK l

2/1

2)(

3.1-13

Motivation Motivation 动机动机MY ENGLISH PROFESSOR once launched into a lecture MY ENGLISH PROFESSOR once launched into a lecture

on "motivation." "What pushes you ahead?" he asked.on "motivation." "What pushes you ahead?" he asked. "What is it that makes you go to school each day? What "What is it that makes you go to school each day? What driving force makes you strive to accomplish?" Turning driving force makes you strive to accomplish?" Turning suddenly to one young woman, he demanded: "What masuddenly to one young woman, he demanded: "What makes you get out of bed in the morning?" The student replkes you get out of bed in the morning?" The student replied: "My mother."ied: "My mother."

我们英文课的教授有一次在课上讲“动机”。“是我们英文课的教授有一次在课上讲“动机”。“是什么推动你在人生的路上向前走？”他问道，“是什么推动你在人生的路上向前走？”他问道，“是什么让你每天上学来？又是什么驱使你追求成什么让你每天上学来？又是什么驱使你追求成功？”冲着一个女学生，他问：“是什么让你早晨功？”冲着一个女学生，他问：“是什么让你早晨从床上爬起来的呢？”学生答道：“我妈妈。”从床上爬起来的呢？”学生答道：“我妈妈。”

The geometry of the step-index circular The geometry of the step-index circular waveguide is shown in Figure 3-1. It waveguide is shown in Figure 3-1. It consists of a core of refractive index nconsists of a core of refractive index n11 and radius a, and a cladding of refractive and radius a, and a cladding of refractive index nindex n22 and radius b. The radius b of the and radius b. The radius b of the cladding is usually chosen to be large cladding is usually chosen to be large enough so that the field of confined modes enough so that the field of confined modes is virtually zero at r=b. In the calculation is virtually zero at r=b. In the calculation below we will put b=∞; this is a legitimate below we will put b=∞; this is a legitimate assumption in most waveguides, as far as assumption in most waveguides, as far as confined modes are concerned.confined modes are concerned.

3.2 the step-index circular waveguide3.2 the step-index circular waveguide

Legitimate 合法的 , 合理的

n(r)

n1

n2

2a

2b

Figure 3-1 Structure and index profile of a step-index circular waveguide

nn11: : core of refractive index ;

n2: cladding of refractive index

a, b: radius. the radius b of the cladding is usually chosen to be large enough so that the field of confined modes is virtually zero at r=b. in the calculation below we will put b=∞ ; this is a legitimate assumption in most waveguides, as far as confined modes are concerned.

The radial dependence of the fields Ez and The radial dependence of the fields Ez and Hz is given by(3.1-10) or (3.1-11), dependinHz is given by(3.1-10) or (3.1-11), depending on the sign of kg on the sign of k22-β-β22. For confined propag. For confined propagation. Β must be larger than nation. Β must be larger than n22ω/c(i.e., β>nω/c(i.e., β>n22kk00=n=n22ω/c). This ensures that the wave is evω/c). This ensures that the wave is evanescent in the cladding region, r>a. The sanescent in the cladding region, r>a. The solution is thus given by (3.1-11) with colution is thus given by (3.1-11) with c11=0. =0. This is evident from the asymptotic behaviThis is evident from the asymptotic behavior for large r given by (3.1-13). The evanescor for large r given by (3.1-13). The evanescent decay of the field also ensures that the ent decay of the field also ensures that the power flow is along the direction of the z apower flow is along the direction of the z axis, i.e., no radial power flow exists. Thus txis, i.e., no radial power flow exists. Thus the fields of a confined mode in the claddihe fields of a confined mode in the cladding (r>a) are given byng (r>a) are given byEvanescent 渐消失的 , 易消散的

The radial dependence of the fields Ez anThe radial dependence of the fields Ez and Hz is given by(3.1-10) or (3.1-11), depend Hz is given by(3.1-10) or (3.1-11), depending on the sign of kding on the sign of k22-β-β22. For confined pr. For confined propagation. Β must be larger than nopagation. Β must be larger than n22ω/c(i.ω/c(i.e., β>ne., β>n22kk00=n=n22ω/c). This ensures that the wω/c). This ensures that the wave is evanescent in the cladding region, ave is evanescent in the cladding region, r>a. The solution is thus given by (3.1-11) r>a. The solution is thus given by (3.1-11) with cwith c11=0. This is evident from the asymp=0. This is evident from the asymptotic behavior for large r given by (3.1-13).totic behavior for large r given by (3.1-13). The evanescent decay of the field also e The evanescent decay of the field also ensures that the power flow is along the dinsures that the power flow is along the direction of the z axis, i.e., no radial power rection of the z axis, i.e., no radial power flow exists. Thus the fields of a confined flow exists. Thus the fields of a confined mode in the cladding (r>a) are given bymode in the cladding (r>a) are given byEvanescent 渐消失的 , 易消散的

asymptotic [ 数 ] 渐近线的 , 渐近的

where C and D are two arbitrary constants , anwhere C and D are two arbitrary constants , and q is given byd q is given by

)](exp[)(),(

)](exp[)(),(

zltiqrDKtrH

zltiqrCKtrE

lz

lz

)](exp[)(),(

)](exp[)(),(

zltiqrDKtrH

zltiqrCKtrE

lz

lz

R>a

(3.2-1)

ck

knq

0

20

22

22

ck

knq

0

20

22

22

3.2-2

For the field in the core, r<a, we must consider oFor the field in the core, r<a, we must consider of the fields as r→0. According to (3.1-12), Yf the fields as r→0. According to (3.1-12), Y l l and and KKl l are divergent as r→0. Since the fields must rare divergent as r→0. Since the fields must remain finite at r=0, the proper choice for the fiemain finite at r=0, the proper choice for the fields in the core (r<a) is (3.1-10) with celds in the core (r<a) is (3.1-10) with c22=0. This =0. This becomes evident only when matching, at the inbecomes evident only when matching, at the interface r=a, the tangential components of the fiterface r=a, the tangential components of the field vectors E and H in the core with the claddineld vectors E and H in the core with the cladding field components derived from (3.2-1); we are g field components derived from (3.2-1); we are unable to accomplish this if the radial dependeunable to accomplish this if the radial dependence of the core fields is given by Ince of the core fields is given by I ll. Thus the pr. Thus the propagation constant βmust be less than nopagation constant βmust be less than n11kk00, an, and the core fields are given byd the core fields are given by

Divergent 分歧的

Tangential 切线的

In the field expressions (3.2-1) and (3.2-3), we In the field expressions (3.2-1) and (3.2-3), we have taken a “+” sign in front of l φ in the have taken a “+” sign in front of l φ in the exponents. A negative sign would yield a set of exponents. A negative sign would yield a set of independent solutions, but with the same radial independent solutions, but with the same radial dependence. Physically, l plays a role similar to dependence. Physically, l plays a role similar to the quantum number describing the z the quantum number describing the z component of the orbital angular momentum of component of the orbital angular momentum of an electron in a cylindrically symmetric an electron in a cylindrically symmetric potential field. Thus, if the positive sign in front potential field. Thus, if the positive sign in front of l φ corresponds to a clockwise “circulation” of l φ corresponds to a clockwise “circulation” of photons about the z axis, the negative of photons about the z axis, the negative

orbital 轨道的 ,potential 潜在的 , 可能的 , 势的 , 位的

sign would corresponds to a clockwise “cirsign would corresponds to a clockwise “circulation” of photons around the axis. Sinculation” of photons around the axis. Since the fiber itself does not possess any prefce the fiber itself does not possess any preferred sense of rotation, these two state are erred sense of rotation, these two state are degenerate.degenerate.

Equations (3.2-1) and (3.2-3) together reqEquations (3.2-1) and (3.2-3) together require that huire that h22>0 and q>0 and q22>0, which translates to>0, which translates to

nn11kk00>β>n>β>n22kk00 (3.2-5) (3.2-5) which can be regarded as a necessary condiwhich can be regarded as a necessary condi

tion for confined modes to exist. This is tion for confined modes to exist. This is

preferred首选的

Identical to the condition discussed in Identical to the condition discussed in Section 13.1 for the slab dielectric Section 13.1 for the slab dielectric waveguide and can be expected on waveguide and can be expected on intuitive grounds from our discussions intuitive grounds from our discussions of total internal reflection at a dielectric of total internal reflection at a dielectric interface. interface.

Using(3.2-1) and (3.2-3) in conjunction Using(3.2-1) and (3.2-3) in conjunction with (3.1-5) and (3.1-6), we can with (3.1-5) and (3.1-6), we can calculate all the field components in calculate all the field components in both the cladding and the core regions. both the cladding and the core regions. The result The result

Slab 厚平板 , 厚片

Intuitive 直觉的Conjunction 联合

The result isThe result is

Boundary 边界 , 分界线

where the primes on Jwhere the primes on Jll and K and Kll again refer to differentia again refer to differentiation with respect to their arguments ha and qa, respecttion with respect to their arguments ha and qa, respectively. Equation(3.2-10) yield a nontrivial solution for ively. Equation(3.2-10) yield a nontrivial solution for

Prime 主要的 , （名词）（数学表达中用来）区别同一变量的不同值

Nontrivial 非平凡的

0)()()(

0)()(

0)()()()(

2'2

2'1

'2

'2

qaKaq

ilDqaK

qChaj

ah

ilBJ

hA

qaDKhaBJ

qaKq

DqaKaq

ilChaJ

hBhaJ

ah

ilA

lllL

ll

lll

0)()()(

0)()(

0)()()()(

2'2

2'1

'2

'2

qaKaq

ilDqaK

qChaj

ah

ilBJ

hA

qaDKhaBJ

qaKq

DqaKaq

ilChaJ

hBhaJ

ah

ilA

lllL

ll

lll

3.2-10

A,B,C and D, provided the determinant of their coefficients vanishes. ThA,B,C and D, provided the determinant of their coefficients vanishes. This requirement yields the following mode condition that determines tis requirement yields the following mode condition that determines the propagation constant he propagation constant

Equation(3.2-11), together with (3.2-4) and (3.2-2), is a transcendental funEquation(3.2-11), together with (3.2-4) and (3.2-2), is a transcendental function of βfor each l. The function J’ction of βfor each l. The function J’ll(x)/xJ(x)/xJll (x) in (3.2-11) is a rapidly v (x) in (3.2-11) is a rapidly varying oscillatory function of x=ha. Therefore, (3.2-11) may be considarying oscillatory function of x=ha. Therefore, (3.2-11) may be considered roughly as a quadratic equation in J’ered roughly as a quadratic equation in J’ll(ha)/haJ(ha)/haJll (ha) . For a given l (ha) . For a given l and a given frequency ω, only a finite number of eigenvalues βcan be and a given frequency ω, only a finite number of eigenvalues βcan be found that satisfy (3.2-11) and (3.2-5). Once the eigenvalues have been found that satisfy (3.2-11) and (3.2-5). Once the eigenvalues have been found, we employ (3.2-10) to solve for the ratios B/A, C/Afound, we employ (3.2-10) to solve for the ratios B/A, C/A

and D/A that determine the six field components of the mode correspoand D/A that determine the six field components of the mode corresponding to each propagation constant β. These ratios are , from (3.2-10),nding to each propagation constant β. These ratios are , from (3.2-10),

Determinant 决定性的eigenvalue] 特征值

New business was opening New business was opening 开业大吉开业大吉

A new business was opening ... and one of the owner's friends wanA new business was opening ... and one of the owner's friends wanted to send him flowers for the occasion. They arrived at the new ted to send him flowers for the occasion. They arrived at the new business site and the owner read the card,.... "Rest in Peace." The business site and the owner read the card,.... "Rest in Peace." The owner was angry and called the florist to complain. After he had towner was angry and called the florist to complain. After he had told the florist of the obvious mistake and how angry he was, the flold the florist of the obvious mistake and how angry he was, the florist replied, "Sir, I'm really sorry for the mistake, but rather thaorist replied, "Sir, I'm really sorry for the mistake, but rather than getting angry, you should imagine this: somewhere, there is a fun getting angry, you should imagine this: somewhere, there is a funeral taking place today, and they have flowers with a note saying,neral taking place today, and they have flowers with a note saying, ... 'Congratulations on your new location!'" ... 'Congratulations on your new location!'"

新公司开业了，开业典礼上，经理的一个朋友送他一个花篮。经新公司开业了，开业典礼上，经理的一个朋友送他一个花篮。经理高声朗读着花篮上的贺卡：“安息吧。”经理生气极了，打电理高声朗读着花篮上的贺卡：“安息吧。”经理生气极了，打电话找来卖花的人要质问他是怎么回事。花店老板来了，看到这个话找来卖花的人要质问他是怎么回事。花店老板来了，看到这个明显的错误和经理气急败坏的样子，他说：“我真得很抱歉。但明显的错误和经理气急败坏的样子，他说：“我真得很抱歉。但是与其这么生气，你倒不如这样想：有另外一个地方，今天要举是与其这么生气，你倒不如这样想：有另外一个地方，今天要举办一个葬礼，他们将会收到一个花篮，留言条上写着‘恭喜你有办一个葬礼，他们将会收到一个花篮，留言条上写着‘恭喜你有了新的归属！’”了新的归属！’”

The radial dependence of the fields Ez and Hz depending on the sign of .

2 2k

For confined propagation, this ensures that the wave is evanescent( 逐渐消失的 ) in the cladding region, r>a. This is evident from the asymptotic( 渐进的 ) behavior for large r given by 3.1-13, the evanescent decay of the field also ensures that the power flow is along the direction of the z axis. i.e., no radial power flow exists.

2 /n c

For the field in the core, r<a, we must consider the behavior of the fields as r → o. And 1 0n k

Schoolwork:

1. Derive the equation 3.1-3 – 3.1-6

2. Translate section 3.2 into Chinese

?

?

??